Rear projection optical system

ABSTRACT

An oblique projection optical system for leading rays of light from a display surface on which an image is displayed to a projection surface in such a way that the ray of light from the center of the display surface is obliquely incident on the projection surface in order to project a magnified image of the image displayed on the display surface onto the projection surface includes a plurality of reflecting surfaces having a power. At least two of the reflecting surfaces have a free-form curved surface, and, of all the reflecting surfaces, the one closest to the projection surface has a negative power and at least one of the other has a positive power. Alternatively, in a rear projection optical system having a projection optical system for projecting an image displayed on a panel display surface onto a screen surface, the projection optical system includes at least four curved-surface reflecting mirrors.

This application is based on Japanese Patent Applications Nos.2001-363653 and 2000-34319 filed on Nov. 29, 2001 and Feb. 7, 2000respectively, the contents of which are hereby incorporated byreference.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a projection optical system forprojecting a magnified image on a screen, and to a rear projectionoptical system provided with such a projection optical system. Morespecifically, the present invention relates to an oblique projectionoptical system and a rear projection optical system that shine a beam oflight on a screen from an oblique direction.

2. Description of the Prior Art

From long ago, it has been common practice to project a magnified imageof an image displayed on a small display surface onto a screen. Before,projection of an image on a screen was generally achieved through frontprojection, whereby the image is projected from in front of the screen,i.e. from the same side as the observer, for example as when a movie isshown in a movie theater. These days, projection of an image is achievedalso through rear projection, whereby the image is projected from behindthe screen by the use of a screen that transmits light. Today,large-screen television sets adopting rear projection are in practicaluse.

It is desired that, except in cases where a large facility like a movietheater itself constitutes a projection apparatus, a projectionapparatus be provided with a large screen and be simultaneously compact.In particular, in a rear projection apparatus that projects an imagefrom behind a screen, it is desired that the apparatus be slim, i.e.that its depth dimension in the direction perpendicular to the screen besmall.

In early models of rear projection apparatus, to make them slim, a verycommon centered optical system is used as a projection optical system,and a flat-surface mirror is arranged behind a screen so as to turn theoptical path of the light exiting from the powered part of theprojection optical system. However, to prevent distortion in the imageformed on the screen, the optical path along the optical axis turned bythe flat-surface mirror needs to run through the center of the screenperpendicularly thereto. This makes it difficult to slim down theapparatus below a certain thickness. The optical path is turnedvertically, because then the turned optical path is shorter than if itis turned horizontally, and usually all the parts, including the displaysurface on which an image is displayed, other than the flat-surfacemirror for turning the optical path are arranged below the screen.

An effective way to further slim down rear projection apparatus is toadopt oblique projection, in which the ray of light striking the centerof the screen, i.e. the ray representing the center of the image, isincident on the screen at a large angle of incidence. However,attempting to achieve oblique projection with a centered projectionoptical system necessitates making the optical path along the opticalaxis turned by the flat-surface mirror run off the center of the screen.Accordingly, the projection optical system needs to include alarge-diameter wide-angle lens of which only part is used forprojection. Such an optical system can be realized, but it entails highcost, and in addition makes the projection optical system itself larger,with little effect of slimming down the apparatus.

To overcome this, proposals have been made to use reflecting mirrorswith curved surfaces as powered elements included in the projectionoptical system. For example, Re—published Patent Application No. WO97/01787 proposes a projection optical system composed of fourcurved-surface mirrors. These curved-surface mirrors have, in order fromthe display surface side, a positive, a negative, a positive, and anegative power. The curved surface closest to the display surface is aspherical surface, and the other three curved surfaces are asphericalsurfaces. The projection optical system that the applicant of thepresent invention proposes in Japanese Patent Application Laid-Open No.2001-221949 also is composed of four curved-surface mirrors. In thisprojection optical system, the curved-surface mirrors have, in orderfrom the display surface side, a positive, a positive, a negative, and anegative power, or a positive, a positive, a negative, and a positivepower. All these surfaces are spherical or aspherical surfaces. Inaddition to these publications, there are more that propose projectionoptical systems composed of three curved-surface mirrors and of othertypes.

Conventionally, an oblique projection optical system composed ofcurved-surface mirrors is, to minimize the lowering of imagingperformance, designed to have a large f-number, and has a long opticalpath length from the display surface on which an image, is displayed tothe projection surface at which a screen is arranged. Moreover, to slimdown the apparatus incorporating it while securing a long optical pathlength, its optical path is turned many times with flat-surface mirrors.The optical path needs to be turned, except on the last occasion, aroundthe screen, specifically below or above the screen, so as not to hinderthe projection of the image on the screen. Thus, an oblique projectionoptical system composed of curved-surface mirrors helps slim down theapparatus incorporating it, but does not contribute to reducing theheight dimension thereof Moreover, in a conventional oblique projectionoptical system, only necessary parts of curved-surface mirrors are usedso as not to hinder miniaturization. Anyway, all these mirrors havespherical or aspherical surfaces that are symmetric about an axis.

As long as a long optical path length is secured to prevent the loweringof imaging performance, it is difficult to reduce the height dimensionof the screen without sacrificing the flatness of the apparatus. Thus,modern oblique projection optical systems are considered to have almostreached the limit in terms of the trade-off between the slimming-down ofprojection apparatus and the reduction of the height dimension thereof.

On the other hand, as described above, rear projection optical systemsused in common rear projection apparatus achieve the slimming-down ofthe apparatus by turning the optical path of the light exiting from aprojection optical system with a single reflecting mirror arrangedbehind a screen. However, the projection optical system used here is ofa centered type, and therefore the ray striking the center of the screensurface needs to be substantially perpendicular to the screen surface.This makes it difficult to slim down rear projection optical systemsbelow a certain thickness.

To overcome this, various optical arrangements have been proposed forfurther slimming-down. For example, Japanese Patent Registered No.2932609 and Japanese Patent Applications Laid-Open No. H3-87731,H2-153338, H2-146535, and H2-130543 disclose rear projection opticalsystems in which the optical path of a projection optical system isturned with two flat-surface reflecting mirrors.

However, with conventional rear projection optical systems, sufficientslimming-down is difficult, or slimming them down poses new problems.For example, the rear projection optical system disclosed in JapanesePatent Registered No. 2932609 mentioned above adopts a method using are-imaging projection optical system in which a displayed image is firstimaged, and the resulting image is then projected on a screen surface soas to be imaged again. This inevitably makes the projection opticalsystem large. In addition, this method requires an oblique projectionoptical system that permits the ray striking the center of the screensurface to be incident thereon at a large angle of incidence, but thepublication describes no specific optical arrangement of such an opticalsystem. The rear projection optical systems disclosed in Japanese PatentApplications Laid-Open No. H3-87731, H2-153338, H2-146535, and H2-130543mentioned above also require oblique projection optical systems forslimming-down, but the publications do not make it clear what specificoptical arrangement to use as a projection optical system.

An oblique projection optical system is usually realized by using partof a centered optical system. However, to slim down a rear projectionoptical system, the projection angle of the principal ray needs to bemade very large. Thus, it is inevitable to use part of a very wide-anglecentered optical system. In general, a wide-angle optical systemrequires a large number of lens elements, and their lens diameters arevery large. This makes the optical system as a whole large.

Some display apparatus incorporate a rear projection optical system thatis slimmed down by actually adopting an oblique projection opticalsystem employing curved-surface reflecting mirrors. In these displayapparatus, however, the light that has exited from the projectionoptical system is reflected directly by a flat-surface reflecting mirrorarranged behind a screen, and thus the curved-surface reflecting mirrorthat constitutes the last surface of the projection optical system needsto be very large. A large curved-surface reflecting mirror like this isdisadvantageous in terms of mass production and cost. Moreover, if theprojection optical system includes only three curved-surface mirrors, itis highly sensitive to errors, and is thus difficult to manufacture.

SUMMARY OF THE INVENTION

An object of the present invention is to provide an oblique projectionoptical system that offers high imaging performance and that permitsfurther miniaturization not only in the direction perpendicular to thescreen but also in the direction along the screen.

Another object of the present invention is to provide a rear projectionoptical system that offers satisfactory optical performance but isnevertheless advantageous in terms of mass production and cost and thatis slim and is composed of compact optical components.

To achieve the above objects, according to one aspect of the presentinvention, an oblique projection optical system for leading rays oflight from a display surface on which an image is displayed to aprojection surface in such a way that the ray of light from the centerof the display surface is obliquely incident on the projection surfacein order to project a magnified image of the image displayed on thedisplay surface onto the projection surface includes a plurality ofreflecting surfaces having a power. Here, at least two of the reflectingsurfaces having a power have a free-form curved surface, and, of all thereflecting surfaces having a power, the one closest to the projectionsurface has a negative power and at least one of the other has apositive power.

According to another aspect of the present invention, in a rearprojection optical system having a projection optical system forprojecting an image displayed on a panel display surface onto a screensurface, the projection optical system includes at least fourcurved-surface reflecting mirrors.

BRIEF DESCRIPTION OF THE DRAWINGS

This and other objects and features of the present invention will becomeclear from the following description, taken in conjunction with thepreferred embodiments with reference to the accompanying drawings inwhich:

FIG. 1 is a sectional view, taken along the x-y plane, of the projectionoptical system of a first embodiment of the invention;

FIG. 2 is a side view, as seen from the z direction, of the projectionoptical system of the first embodiment;

FIG. 3 is a top view, as seen from the y direction, of the projectionoptical system of the first embodiment;

FIG. 4 is a front view, as seen from the x direction, of the projectionoptical system of the first embodiment;

FIG. 5 is a spot diagram obtained on the projection surface of theprojection optical system of the first embodiment;

FIG. 6 is a diagram showing the distortion observed on the projectionsurface of the projection optical system of the first embodiment,

FIG. 7 is a sectional view, taken along the x-y plane, of the projectionoptical system of a modified example of the first embodiment;

FIG. 8 is a top view, as seen from the y direction, of the projectionoptical system of the modified example of the first embodiment;

FIG. 9 is a sectional view, taken along the x-y plane, of the projectionoptical system of another modified example of the first embodiment;

FIG. 10 is a sectional view, taken along the x-y plane, of theprojection optical system of a second embodiment of the invention;

FIG. 11 is a side view, as seen from the z direction, of the projectionoptical system of the second embodiment;

FIG. 12 is a top view, as seen from the y direction, of the projectionoptical system of the second embodiment;

FIG. 13 is a spot diagram obtained on the projection surface of theprojection optical system of the second embodiment;

FIG. 14 is a diagram showing the distortion observed on the projectionsurface of the projection optical system of the second embodiment;

FIG. 15 is a sectional view, taken along the x-y plane, of theprojection optical system of a third embodiment of the invention;

FIG. 16 is a side view, as seen from the z direction, of the projectionoptical system of the third embodiment;

FIG. 17 is a top view, as seen from the y direction, of the projectionoptical system of the third embodiment;

FIG. 18 is a spot diagram obtained on the projection surface of theprojection optical system of the third embodiment;

FIG. 19 is a diagram showing the distortion observed on the projectionsurface of the projection optical system of the third embodiment;

FIG. 20 is a sectional view, taken along the x-y plane, of theprojection optical system of a fourth embodiment of the invention;

FIG. 21 is a side view, as seen from the z direction, of the projectionoptical system of the fourth embodiment;

FIG. 22 is a top view, as seen from the y direction, of the projectionoptical system of the fourth embodiment;

FIG. 23 is a spot diagram obtained on the projection surface of theprojection optical system of the fourth embodiment;

FIG. 24 is a diagram showing the distortion observed on the projectionsurface of the projection optical system of the fourth embodiment;

FIG. 25 is a sectional view, taken along the x-y plane, of theprojection optical system of a fifth embodiment of the invention;

FIG. 26 is a side view, as seen from the z direction, of the projectionoptical system of the fifth embodiment;

FIG. 27 is a top view, as seen from the y direction, of the projectionoptical system of the fifth embodiment;

FIG. 28 is a spot diagram obtained on the projection surface of theprojection optical system of the fifth embodiment;

FIG. 29 is a diagram showing the distortion observed on the projectionsurface of the projection optical system of the fifth embodiment;

FIG. 30 is a sectional view, taken along the x-y plane, of theprojection optical system of a sixth embodiment of the invention;

FIG. 31 is a side view, as seen from the z direction, of the projectionoptical system of the sixth embodiment;

FIG. 32 is a top view, as seen from the y direction, of the projectionoptical system of the sixth embodiment;

FIG. 33 is a spot diagram obtained on the projection surface of theprojection optical system of the sixth embodiment;

FIG. 34 is a diagram showing the distortion observed on the projectionsurface of the projection optical system of the sixth embodiment;

FIG. 35 is a sectional view, taken along the x-y plane, of theprojection optical system of a seventh embodiment of the invention;

FIG. 36 is a side view, as seen from the z direction, of the projectionoptical system of the seventh embodiment;

FIG. 37 is a top view, as seen from the y direction, of the projectionoptical system of the seventh embodiment;

FIG. 38 is a spot diagram obtained on the projection surface of theprojection optical system of the seventh embodiment;

FIG. 39 is a diagram showing the distortion observed on the projectionsurface of the projection optical system of the seventh embodiment;

FIG. 40 is an optical path diagram of the rear projection optical systemof an eighth embodiment of the invention;

FIG. 41 is a diagram showing the projection optical system constitutingthe eighth embodiment and a principal portion of the optical paththereof;

FIG. 42 is a spot diagram of the eighth embodiment;

FIG. 43 is a distortion diagram of the eighth embodiment;

FIG. 44 is an optical path diagram of the rear projection optical systemof a ninth embodiment of the invention;

FIG. 45 is a diagram showing the projection optical system constitutingthe ninth embodiment and a principal portion of the optical paththereof;

FIG. 46 is a spot diagram of the ninth embodiment;

FIG. 47 is a distortion diagram of the ninth embodiment;

FIG. 48 is an optical path diagram of the rear projection optical systemof a tenth embodiment of the invention;

FIG. 49 is a diagram showing the projection optical system constitutingthe tenth embodiment and a principal portion of the optical paththereof;

FIG. 50 is a spot diagram of the tenth embodiment;

FIG. 51 is a distortion diagram of the tenth embodiment; and

FIG. 52 is a diagram showing the structure of a principal portion of ascreen suitable for use in the eighth to tenth embodiments and theoptical path therethrough.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

Hereinafter, embodiments of the present invention will be described withreference to the accompanying drawings and tables.

First to Seventh Embodiments

First, as a first to a seventh embodiment of the invention, practicalexamples (Examples 1 to 7, respectively) of oblique projection opticalsystems will be presented below with reference to their actualconstruction data and other data. The oblique projection optical systems1 to 7 of Examples 1 to 7 are all composed of four poweredcurved-surface mirrors and one non-powered flat-surface mirror. Theseoblique projection optical systems 1 to 7 are all designed to lead raysof light from a rectangular display surface having longer sides in thewidth direction thereof to a projection surface by reflecting the rayswith the individual mirrors in such a way as to deflect the rays in theheight direction of the display surface and make the rays converge onthe projection surface. As a result, a magnified image of the imagedisplayed on the display surface is formed (projected) on the projectionsurface, in a rectangular area thereon that is substantially similar tothe display surface.

In each example, the display surface is represented by S0, and theprojection surface is represented by S6. The reflecting surfaces of theindividual mirrors are represented by S1 to S5 in the order in whichthey receive light from the display surface S0 (i.e. in order ofproximity to the display surface S0 along the optical path). The pupilplane (aperture stop) is represented by APR.

The oblique projection optical systems 1 to 7 of Examples 1 to 7include, as all or part of the powered reflecting surfaces S1 to S4,free-form curved surfaces. Thus, any of these optical systems issymmetric about a plane, but has no optical axis that holds throughoutthe optical system. Accordingly, it is not proper to define the surfacesS0 to S6 in a coordinate system that uses an optical axis as areference. Instead, in an absolute coordinate system, local coordinatesystems are defined one for each of the surface S0 to S6, so that thesurfaces S0 to S6 are represented by their respective coordinatesystems. Then, the optical system as a whole is defined in terms of thepositions and directions of the individual local coordinate systems inthe absolute coordinate system.

Here, Cartesian coordinate systems are used as the absolute and localcoordinate systems. The coordinate axes of the absolute coordinatesystems are referred to as the x-axis, y-axis, and z-axis, and thecoordinate axes of a local coordinate system are referred to as theX-axis, Y-axis, and Z-axis. All lengths are given in mm. The absolutecoordinate system has its origin at the center of the display surfaceS0, and has its x-, y-, and z-axes respectively in the direction normalto the projection surface S6, in the height direction thereof, and inthe width direction thereof. Each local coordinate system has its originon the x-y plane of the absolute coordinate system. For each localcoordinate system, the unit vectors along its X-, Y-, and Z-axes arerepresented respectively by VX, VY, and VZ, and these unit vectors VX,VY, and VZ are defined in the absolute coordinate system to define thedirection of the local coordinate system in the absolute coordinatesystem. The Z-axis of each local coordinate system is parallel to thez-axis of the absolute coordinate system, and therefore the X-Y planecoincides with the x-y plane. The surfaces S0 to S6 are symmetric aboutthe X-Y plane of their respective local coordinate systems, and theoptical system as a whole is symmetric about the x-y plane of theabsolute coordinate system.

The surfaces S0 to S6 are each defined by the formula of an extendedaspherical surface ES below. $\begin{matrix}{X = {\frac{{C0} \cdot H^{2}}{1 + \left( {1 - {ɛ \cdot {C0}^{2} \cdot H^{2}}} \right)^{1/2}} + {\sum\limits_{i}{{Ai} \cdot H^{1}}} + {\sum\limits_{j.k}{{Bjk} \cdot Y^{j} \cdot Z^{k}}}}} & ({ES})\end{matrix}$

In this formula, C0 represents the curvature at the vertex (theintersection with the X-axis); ε represents the conic constant; i, j,and k each represent an integer number equal to or greater than 0; andH²=Y²+Z². Ai represents the coefficient for the term that includes H tothe power of i, and Bjk represents the coefficient for the term thatincludes Y to the power of j and Z to the power of k. In each example,terms including H to the power i of up to 12 are considered, and termsincluding Y and Z to the power j+k of 10 are considered. In thepresentation of each example, the values of the coefficients Ai and Bjkare given, but those of which the value equals 0 are omitted unlessnecessary.

EXAMPLE 1

FIGS. 1 to 4 show the construction and optical path of the obliqueprojection optical system 1 of Example 1, and Tables 3 to 10 show theconstruction data thereof Tables 3 to 10 each list the data of thesurface referred to by the symbol noted at the top of the table. N0 andN1 respectively represent the refractive indices of the media beforeincidence and after incidence (i.e. after reflection) on a surface.“Position” indicates the position of the origin of the correspondinglocal coordinate system in the absolute coordinate system. In Table 5,which lists the data of the pupil plane APR, R represents the radius ofthe pupil (aperture stop).

It is to be noted that, also in the other examples described later, whattheir construction data represents is the same as with Tables 3 to 10.

FIG. 1 is a sectional view taken along the x-y plane, and shows thesurfaces S0 to S6 together with, among the rays emanating from thecenter in the width direction of the display surface S0, five rays, i.e.two emanating from both ends (end points) in the height direction of thedisplay surface S0 and three emanating from the three points that dividethe line between those ends into four equal parts. FIG. 2 is a side viewas seen from the z direction, and shows, in a form superposed on thefive rays mentioned above, among the rays emanating from both ends inthe width direction of the display surface S0, those emanating from thesame points in the height direction of the display surface S0 asdescribed above. Here, the surfaces are not marked with their symbols S0to S6.

FIG. 3 is a top view as seen from the y direction, and shows thesurfaces S0 to S6 together with, among the rays emanating from thecenter in the height direction of the display surface S0, nine rays,i.e. two emanating from both ends in the width direction of the displaysurface S0 and seven emanating from the seven points that divide theline between those ends into eight equal parts. FIG. 4 is a front viewas seen from the x direction, and shows the surfaces S0 to S6 togetherwith, among the rays emanating from both ends (end lines) in the heightdirection of the display surface S0 and from three lines that divide thearea between those ends into four equal parts, 45 rays in total, i.e.,for each of these five lines, two emanating from both ends in the widthdirection of the display surface S0 and seven emanating from the sevenpoints that divide the line between those ends into eight equal parts.

FIG. 5 shows a spot diagram obtained on the projection surface S6, morespecifically, near the intersections between, on the one hand, both ends(end lines) in the height direction of the projection surface S6 and thelines that divide the area between those ends into four equal parts and,on the other hand, both ends (end lines) in the width direction of theprojection surface S6 and the lines that divide the area between thoseends into eight equal parts. These intersections correspond to theorigins of the coordinate axes shown in the individual sections of thediagram. Since the optical system is symmetric in the width direction(the z direction), the obtained results are shown for only half of theprojection surface S6. That is, FIG. 5 is a diagram for 25 among the 45rays shown in FIG. 4, and the third-from-the-above, leftmost section ofthe diagram shows the results obtained near the center of the projectionsurface S6. In each section of the diagram, spots represent thepositions in which different rays belonging to an identical beam oflight are incident. Above each section of the diagram are noted thecoordinates (in the local coordinate system) of the center of theincident positions of all the rays belonging to an identical beam. Thevalues ±2 noted by the coordinate axes in each section of the diagramrepresent the distances from the origin of those coordinate axes.

FIG. 6 shows the distortion of the image observed on the projectionsurface S6. In this figure, solid lines represent the results obtainedwith the oblique projection optical system 1, and broken lines representthe ideal result without distortion.

It is to be noted that, also in the modified examples and the otherexamples described later, what their sectional view, top view, sideview, spot diagram, and distortion diagram represent is the same as withFIGS. 1 to 3, 5, and 6.

As shown in Tables 4 to 8, in the oblique projection optical system 1 ofthis example, the reflecting surface S1 is a spherical surface, thereflecting surface S2 is an aspherical surface, the reflecting surfaceS3 is a free-form curved surface, and the reflecting surface S4 is afree-form curved surface. The pupil plane APR is located between thereflecting surface S1 and the reflecting surface S2. As shown in Tables3, 9, and 10, the display surface S0, the reflecting surface S5, and theprojection surface S6 are flat surfaces, which are parallel to oneanother.

Table 1 shows the magnifications factors (the projection magnificationfactors) by which an image is magnified when projected, the sizes of thedisplay surface S0 and the projection surface S6 (i.e. the sizes of theareas in which an image is displayed or projected), the angles ofincidence at which rays are incident on the projection surface S6, andthe position of the entrance pupil of the beam from the display surfaceS0 as observed in this example, together with the same data as observedin the other examples. Here, the sizes of the display surface S0 and theprojection surface S6 are given in half values. The position of theentrance pupil is given as values of X and Y in the local coordinatesystem of the display surface S0, with Y given as a length when X has afinite value, and with Y given as an angle when X is infinite.

The magnification factors are those obtained from the center of thedisplay surface S0 to the center of the projection surface S6, with β(Y)representing the magnification factor in the height direction (the Y,and thus y, direction), and with β(Z) representing the magnificationfactor in the width direction (the Z, and thus z, direction). Themagnification factor β calculated as the ratio of the size of theprojection surface S6 to the size of the display surface S0 isapproximately equal to those listed in the table. The magnificationfactors β(Y) and β(Z) in the height and width directions are almostequal to each other, and the slight difference between them is givenunder “Anamo(Y/Z).” It is to be noted that the reason that themagnification factor β(Z) in the width direction takes a negative valueis that the Z-axis of the local coordinate system points in the oppositedirection from one of the reflecting surfaces S1 to S5 to the next.

In the oblique projection optical system 1, the magnification factorsβ(Y) and β(Z) in the height and width directions are 78.57 and 78.56respectively. Rays are incident on the projection surface S6 at theminimum angle of incidence (24.0°) at the lower end in the heightdirection at the center in the width direction, at the maximum angle ofincidence (67.3°) at the upper end in the height direction at both endsin the width direction, and at an angle of incidence of 52.1° at thecenter. Thus, the angle of view, which is defined as the differencebetween the maximum and minimum angles of incidence, is 43.3°. Theentrance pupil is located at infinity, making the optical system anoblique telecentric optical system.

Table 2 shows the f-numbers, the thickness D of the optical path, thelength H in the height direction of the projection surface S6, the ratioof the thickness D of the optical path to the length H of the projectionsurface S6, the shapes of the powered reflecting surfaces S1 to S4, andthe signs of their powers as observed in this example, together with thesame data as observed in the other example. Here, the f-numbers areeffective f-numbers calculated from the diameter and position of theentrance pupil. Fnoy represents the f-number in the height direction ofthe display surface S0, and Fnoz represents the f-number in the widthdirection thereof The thickness D of the optical path is the maximumlength, in the direction normal to the projection surface S6 (i.e. inthe X and thus x, direction), of the space through which light travelsfrom the display surface S0 to projection surface S6.

The symbols “sp,” “asp,” and “exasp” respectively denote spherical,aspherical, and free-form curved surfaces. The symbol (+) denotes aconcave surface having a positive power, and the symbol (−) denotes aconvex surface having a negative power. Here, the powers of thereflecting surfaces S1 to S4 depend on the surface shape thereof at thepoint at which the ray that travels from the center of the displaysurface S0 through the center of the pupil plane APR to the projectionsurface S6 passes therethrough, and not on the sign of the curvature C0in formula (ES) noted earlier by which the curved surfaces are defined.

In the oblique projection optical system 1, the f-numbers Fnoy and Fnozin the height and width directions are 3.5 and 3.4 respectively, and theratio D/H of the thickness of the optical path to the length in theheight direction of the projection surface S6 is 0.30. The fourreflecting surfaces S1 to S4 have, from the display surface S0 side, apositive, a negative, a positive, and a negative power. Thus, thereflecting surface S4 closest to the projection surface S6 has anegative power.

Whereas the length H in the height direction of the projection surfaceS6 is about 685 mm, the length in the height direction of the part ofthe optical system that is located below the lower end of the projectionsurface S6 is about 210 mm, which is about 23% of the length in theheight direction of the optical system as a whole. The distance betweenthe projection surface S6 and the flat-surface reflecting surface S5 is200 mm, which is almost equal to the thickness D of the optical path.The ratio of the length of the projection surface S6 in the heightdirection to that in the width direction is about 9:16.

MODIFIED EXAMPLE 1 OF EXAMPLE 1

FIGS. 7 and 8 show a sectional view and a top view, respectively, of theoblique projection optical system 1-b of a modified example of theoblique projection optical system 1 of Example 1. The oblique projectionoptical system 1-b differs from its base model in that the surfacesstarting with the display surface S0 and ending with the flat-surfacereflecting surface S5 are shifted toward the projection surface S6.

As will be clear from FIG. 7, the greater part of the reflecting surfaceS4 is located opposite to the flat-surface reflecting surface S5 withrespect to the projection surface S6, and thus a lower central portionof the oblique projection optical system 1-b protrudes a little from theprojection surface S6. As a result, as shown in Table 2 describedearlier, the thickness D of the optical path in the optical system as awhole is larger, but, as will be understood through comparison betweenFIGS. 7 and 1, the distance between the projection surface S6 and theflat-surface reflecting surface S5 is shorter than in the obliqueprojection optical system 1.

MODIFIED EXAMPLE 2 OF EXAMPLE 1

FIG. 9 shows a sectional view of the oblique projection optical system1-c of another modified example of the oblique projection optical system1 of Example 1. This oblique projection optical system 1-c differs fromits base model in that the flat-surface reflecting surface S5 isomitted. As a result of the omission of the reflecting surface S5, asshown in Table 2, the thickness D of the optical path in the opticalsystem as a whole doubles. Still, the thickness D of the optical path isabout 59% of the length H in the height direction of the projectionsurface S6.

Whereas the oblique projection optical system 1 provided with thereflecting surface S5 is suitable only for rear projection, the obliqueprojection optical system 1-c is suitable for both rear projection andfront projection. It is to be understood that, in the oblique projectionoptical systems of the examples described below, the reflecting surfaceS5 may be omitted as in this modified example.

EXAMPLE 2

FIGS. 10 to 12 show a sectional view, a side view, and a top view,respectively, of the oblique projection optical system 2 of Example 2,and Tables 11 to 18 show the construction data thereof FIG. 13 shows aspot diagram obtained on the projection surface S6, and FIG. 14 showsthe distortion of the image observed on the projection surface S6.

As shown in Table 1, the magnification factors β(Y) and β(Z) in theheight and width directions are 71.41 and 71.39 respectively. Rays areincident on the projection surface S6 at the minimum angle of incidence(24.7°) at the lower end in the height direction at the center in thewidth direction, at the maximum angle of incidence (65.8°) at the upperend in the height direction at both ends in the width direction, and atan angle of incidence of 50.9° at the center. Thus, the angle of view is41.1°. The entrance pupil is located at infinity on the normal to thecenter of the display surface S0, making the optical system atelecentric optical system.

As shown in Table 2, the f-numbers Fnoy and Fnoz in the height and widthdirections are 2.5 and 2.5 respectively, and the ratio D/H of thethickness of the optical path to the length in the height direction ofthe projection surface S6 is 0.32. The four reflecting surfaces S1 to S4have, from the display surface S0 side, a positive, a negative, apositive, and a negative power. Thus, the reflecting surface S4 closestto the projection surface S6 has a negative power. The two reflectingsurfaces S1 and S2 close to the display surface S0 are asphericalsurfaces, and the two reflecting surfaces S3 and S4 close to theprojection surface S6 are free-form curved surfaces.

Whereas the length H in the height direction of the projection surfaceS6 is about 623 mm, the length in the height direction of the part ofthe optical system that is located below the lower end of the projectionsurface S6 is about 233 mm, which is about 27% of the length in theheight direction of the optical system as a whole. The distance betweenthe projection surface S6 and the flat-surface reflecting surface S5 is200 mm, which is almost equal to the thickness D of the optical path.The ratio of the length of the projection surface S6 in the heightdirection to that in the width direction is about 9:16.

EXAMPLE 3

FIGS. 15 to 17 show a sectional view, a side view, and a top view,respectively, of the oblique projection optical system 3 of Example 3,and Tables 19 to 26 show the construction data thereof FIG. 18 shows aspot diagram obtained on the projection surface S6, and FIG. 19 showsthe distortion of the image observed on the projection surface S6.

As shown in Table 1, the magnification factors β(Y) and β(Z) in theheight and width directions are 71.40 and 71.39 respectively. Rays areincident on the projection surface S6 at the minimum angle of incidence(24.3°) at the lower end in the height direction at the center in thewidth direction, at the maximum angle of incidence (65.7°) at the upperend in the height direction at both ends in the width direction, and atan angle of incidence of 50.7° at the center. Thus, the angle of view is41.4°. The entrance pupil is located at infinity, making the opticalsystem an oblique telecentric optical system.

As shown in Table 2, the f-numbers Fnoy and Fnoz in the height and widthdirections are 2.6 and 2.5 respectively, and the ratio D/H of thethickness of the optical path to the length in the height direction ofthe projection surface S6 is 0.32. The four reflecting surfaces S1 to S4have, from the display surface S0 side, a positive, a negative, apositive, and a negative power. Thus, the reflecting surface S4 closestto the projection surface S6 has a negative power. The two reflectingsurfaces S1 and S2 close to the display surface S0 are asphericalsurfaces, and the two reflecting surfaces S3 and S4 close to theprojection surface S6 are free-form curved surfaces.

Whereas the length H in the height direction of the projection surfaceS6 is about 623 mm, the length in the height direction of the part ofthe optical system that is located below the lower end of the projectionsurface S6 is about 225 mm, which is about 27% of the length in theheight direction of the optical system as a whole. The distance betweenthe projection surface S6 and the flat-surface reflecting surface S5 is200 mm, which is almost equal to the thickness D of the optical path.The ratio of the length of the projection surface S6 in the heightdirection to that in the width direction is about 9:16.

EXAMPLE 4

FIGS. 20 to 22 show a sectional view, a side view, and a top view,respectively, of the oblique projection optical system 4 of Example 4,and Tables 27 to 34 show the construction data thereof FIG. 23 shows aspot diagram obtained on the projection surface S6, and FIG. 24 showsthe distortion of the image observed on the projection surface S6.

The pupil plane APR is located between the reflecting surface S1 closestto the display surface S0 and the display surface S0, making the obliqueprojection optical system 4 a rear-aperture-type optical system. Theaperture stop may be located between the reflecting surfaces S1 and S2so as not to obstruct the beam traveling therebetween.

As shown in Table 1, the magnification factors β(Y) and β(Z) in theheight and width directions are 51.23 and 51.27 respectively. Rays areincident on the projection surface S6 at the minimum angle of incidence(35.9°) at the lower end in the height direction at the center in thewidth direction, at the maximum angle of incidence (71.4°) at the upperend in the height direction at both ends in the width direction, and atan angle of incidence of 61.3° at the center. Thus, the angle of view is35.5°. The entrance pupil is located at a finite distance from thedisplay surface S0, making the optical system a non-telecentric opticalsystem.

As shown in Table 2, the f-numbers Fnoy and Fnoz in the height and widthdirections are 3.6 and 3.5 respectively, and the ratio D/H of thethickness of the optical path to the length in the height direction ofthe projection surface S6 is 0.26. The four reflecting surfaces S1 to S4have, from the display surface S0 side, a positive, a negative, apositive, and a negative power. Thus, the reflecting surface S4 closestto the projection surface S6 has a negative power. All these fourreflecting surfaces S1 to S4 are free-form curved surfaces.

Whereas the length H in the height direction of the projection surfaceS6 is about 498 mm, the length in the height direction of the part ofthe optical system that is located below the lower end of the projectionsurface S6 is about 228 mm, which is about 31% of the length in theheight direction of the optical system as a whole. The distance betweenthe projection surface S6 and the flat-surface reflecting surface S5 is110 mm, which is about 31 mm less than the thickness D of the opticalpath. Thus, a lower central portion of the optical system protrudes alittle from the projection surface S6. The ratio of the length of theprojection surface S6 in the height direction to that in the widthdirection is about 9:16.

In the oblique projection optical systems 1 to 3 of Examples 1 to 3described earlier, the beam traveling from the display surface S0 to thereflecting surface S1 has a high degree of symmetry in the heightdirection of the display surface S0. Therefore, it is difficult toilluminate the panel that displays the image on the display surface S0from the reflecting surface S1 side. Thus, the oblique projectionoptical systems 1 to 3 are suitable for use with a transmissive imagedisplay panel, such as a transmissive liquid crystal panel, is used.

By contrast, in the oblique projection optical system 4 of this example,the beam traveling from the display surface S0 to the reflecting surfaceS1 shows striking asymmetry in the height direction of the displaysurface S0. Therefore, it is possible to illuminate the image displaypanel from the reflecting surface S1 side. Thus, the oblique projectionoptical system 4 is suitable for use with both a transmissive panel anda reflective panel. As a reflective panel, it is possible to use areflective liquid crystal panel, or a mirror device composed of a largenumber of minute mirror elements which modulates illumination light byvarying the direction of the individual mirror elements.

EXAMPLE 5

FIGS. 25 to 27 show a sectional view, a side view, and a top view,respectively, of the oblique projection optical system 5 of Example 5,and Tables 35 to 42 show the construction data thereof FIG. 28 shows aspot diagram obtained on the projection surface S6, and FIG. 29 showsthe distortion of the image observed on the projection surface S6.

As shown in Table 1, the magnification factors β(Y) and β(Z) in theheight and width directions are 51.25 and 51.27 respectively. Rays areincident on the projection surface S6 at the minimum angle of incidence(34.8°) at the lower end in the height direction at the center in thewidth direction, at the maximum angle of incidence (69.7°) at the upperend in the height direction at both ends in the width direction, and atan angle of incidence of 59.8° at the center. Thus, the angle of view is34.9°. The entrance pupil is located at a finite distance from thedisplay surface S0, making the optical system a non-telecentric opticalsystem.

As shown in Table 2, the f-numbers Fnoy and Fnoz in the height and widthdirections are 3.7 and 3.5 respectively, and the ratio D/H of thethickness of the optical path to the length in the height direction ofthe projection surface S6 is 0.24. The four reflecting surfaces S1 to S4have, from the display surface S0 side, a positive, a negative, apositive, and a negative power. Thus, the reflecting surface S4 closestto the projection surface S6 has a negative power. All these fourreflecting surfaces S1 to S4 are free-form curved surfaces.

Whereas the length H in the height direction of the projection surfaceS6 is about 498 mm, the length in the height direction of the part ofthe optical system that is located below the lower end of the projectionsurface S6 is about 234 mm, which is about 32% of the length in theheight direction of the optical system as a whole. The distance betweenthe projection surface S6 and the flat-surface reflecting surface S5 is120 mm, which is equal to the thickness D of the optical path. The ratioof the length of the projection surface S6 in the height direction tothat in the width direction is about 9:16.

EXAMPLE 6

FIGS. 30 to 32 show a sectional view, a side view, and a top view,respectively, of the oblique projection optical system 6 of Example 6,and Tables 43 to 50 show the construction data thereof FIG. 33 shows aspot diagram obtained on the projection surface S6, and FIG. 34 showsthe distortion of the image observed on the projection surface S6.

As shown in Table 1, the magnification factors β(Y) and β(Z) in theheight and width directions are 70.45 and 70.50 respectively. Rays areincident on the projection surface S6 at the minimum angle of incidence(40.2°) at the lower end in the height direction at the center in thewidth direction, at the maximum angle of incidence (73.2°) at the upperend in the height direction at both ends in the width direction, and atan angle of incidence of 64.3° at the center. Thus, the angle of view is33.0°. The entrance pupil is located very far away from the displaysurface S0, making the optical system a non-telecentric optical systemclose to a telecentric optical system.

As shown in Table 2, the f-numbers Fnoy and Fnoz in the height and widthdirections are 3.4 and 3.3 respectively, and the ratio D/H of thethickness of the optical path to the length in the height direction ofthe projection surface S6 is 0.21. The four reflecting surfaces S1 to S4have, from the display surface S0 side, a positive, a negative, apositive, and a negative power. Thus, the reflecting surface S4 closestto the projection surface S6 has a negative power. All these fourreflecting surfaces S1 to S4 are free-form curved surfaces.

Whereas the length H in the height direction of the projection surfaceS6 is about 685 mm, the length in the height direction of the part ofthe optical system that is located below the lower end of the projectionsurface S6 is about 289 mm, which is about 30% of the length in theheight direction of the optical system as a whole. The distance betweenthe projection surface S6 and the flat-surface reflecting surface S5 is145 mm, which is equal to the thickness D of the optical path. The ratioof the length of the projection surface S6 in the height direction tothat in the width direction is about 9:16.

EXAMPLE 7

FIGS. 35 to 37 show a sectional view, a side view, and a top view,respectively, of the oblique projection optical system 7 of Example 7,and Tables 51 to 58 show the construction data thereof FIG. 38 shows aspot diagram obtained on the projection surface S6, and FIG. 39 showsthe distortion of the image observed on the projection surface S6.

As shown in Table 1, the magnification factors β(Y) and β(Z) in theheight and width directions are 51.24 and 51.27 respectively. Rays areincident on the projection surface S6 at the minimum angle of incidence(36.8°) at the lower end in the height direction at the center in thewidth direction, at the maximum angle of incidence (69.1°) at the upperend in the height direction at both ends in the width direction, and atan angle of incidence of 58.6° at the center. Thus, the angle of view is32.2°. The entrance pupil is located at infinity, making the opticalsystem a telecentric optical system.

As shown in Table 2, the f-numbers Fnoy and Fnoz in the height and widthdirections are both 2.5, and the ratio D/H of the thickness of theoptical path to the length in the height direction of the projectionsurface S6 is 0.31. The four reflecting surfaces S1 to S4 have, from thedisplay surface S0 side, a positive, a negative, a positive, and anegative power. Thus, the reflecting surface S4 closest to theprojection surface S6 has a negative power. All these four reflectingsurfaces S1 to S4 are free-form curved surfaces.

Whereas the length H in the height direction of the projection surfaceS6 is about 498 mm, the length in the height direction of the part ofthe optical system that is located below the lower end of the projectionsurface S6 is about 233 mm, which is about 32% of the length in theheight direction of the optical system as a whole. The distance betweenthe projection surface S6 and the flat-surface reflecting surface S5 is125 mm, which is about 27 mm less than the thickness D of the opticalpath. Thus, a lower central portion of the optical system protrudes alittle from the projection surface S6. The ratio of the length of theprojection surface S6 in the height direction to that in the widthdirection is about 9:16.

In Examples 1 to 7 described thus far, the display surface S0 and theprojection surface S6 are arranged parallel. However, the displaysurface S0 may be arranged so as to be inclined relative to theprojection surface S6. Such arrangement is easy in the obliqueprojection optical systems 1 to 7 provided with reflecting surfaceshaving free-form curved surfaces. In these examples, all the poweredsurfaces are reflecting surfaces. However, part of the powered surfacesmay be realized with refractive surfaces. That is, in the obliqueprojection optical systems 1 to 7, it is possible to use lenses incombination with mirrors, or to use lenses instead of the mirrors havingcurved or aspherical surfaces.

More than one display surface may be provided; that is, it is possible,by the use of a cross prism or the like, to provide a plurality ofdisplay surfaces that are optically equivalent to one another. Forexample, by arranging a cross dichroic prism between the display surfaceS0 and the reflecting surface S1, it is possible to arrange two displaysurfaces equivalent to the display surface S0. Then, by displaying red,green, and blue components of an image on these three display surfaces,and then integrating together the light of these color components withthe cross dichroic prism, it is possible to form a color image on theprojection surface S6. In any of the oblique projection optical systems1 to 7, there is sufficient room to arrange such a cross prism in aportion of the space between the reflecting surface S1 and the displaysurface S0 close to the display surface S0. It is to be noted that, evenwith a single display surface S0, it is possible to present a colorimage by displaying red, green, and blue components of an image thereonon a time division basis.

As described above, in the first to seventh embodiments, in an obliqueprojection optical system that projects a magnified image of an imagedisplayed on a display surface onto a projection surface and that isprovided with a plurality of reflecting surfaces having a power, therays of light from the display surface on which the image is displayedare led to the projection surface in such a way that the ray from thecenter of the display surface is obliquely incident on the projectionsurface. At least two of the reflecting surfaces having a power arefree-form curved surfaces. Moreover, of the reflecting surfaces having apower, the one closest to the projection surface has a negative power,and at least one of the other reflecting surfaces has a positive power.

This oblique projection optical system, as opposed to conventionalprojection optical systems built as centered optical systems, adoptsfree-form curved surfaces as reflecting surfaces. Adopting free-formcurved surfaces as reflecting surfaces makes it possible to achieveoblique projection almost free from distortion with a short optical pathlength without sacrificing imaging performance. By arranging a means fordisplaying an image on the display surface and a screen on theprojection surface, it is possible to obtain a projection apparatus.Having a short optical path length, the projection apparatus thusobtained is not only slim, i.e. has a small dimension in the directionperpendicular to the screen, but has a small dimension also in thedirection along the screen. When provided with a flat-surface mirror forturning the optical path, this oblique projection optical system issuitable for use in rear projection apparatus, but it can be used infront projection apparatus as well.

To obtain a sufficiently high magnification factor (projectionmagnification factor), it is preferable that, of the reflecting surfaceshaving a power, the one closest to the projection surface have anegative power. Accordingly, to permit the rays from different points onthe display surface to converge on one point on the projecting surface,at least one of the other reflecting surfaces having a power needs tohave a positive power. Thus, the powers of the reflecting surfaces aredetermined so as to fulfill these requirements.

The oblique projection optical system described above has fourreflecting surfaces having a power, and it is preferable that thesereflecting surfaces have a positive, a negative, a positive, and anegative power in order of proximity to the display surface. This makesit easy to shorten the optical path from the display surface to theprojection surface, and thus to slim down a projection apparatusincorporating the oblique projection optical system while simultaneouslyreducing the dimension of the projection apparatus in the directionalong the screen.

The oblique projection optical system may be so configured that thedisplay surface has a smaller dimension in the height direction than inthe width direction, that the reflecting surfaces having a power eachreflect the rays of light from the display surface in such a way as todeflect the rays in the height direction of the display surface, thatthe pupil plane is located between the one of the reflecting surfaceshaving a power that is second-closest to the display surface and thedisplay surface, and that the following conditions are fulfilled:Fnoy≧Fnoz, Fnoy≦4.5, and Fnoz≦4.0, where Fnoy represents the f-number inthe direction corresponding to the height direction of the displaysurface, and Fnoz represents the f-number in the direction correspondingto the width direction of the display surface. Making each reflectingsurface reflect the rays from the display surface in such a way as todeflect the rays in the height direction of the display surface makes iteasy to reduce the size of a projection apparatus incorporating theoblique projection optical system in the height direction of the screen.Moreover, setting the position of the pupil plane and the relationshipbetween the f-numbers in this way make it possible to present brightimages.

Alternatively, the oblique projection optical system may be soconfigured that the display surface has a smaller dimension in theheight direction than in the width direction, the reflecting surfaceshaving a power each reflect the rays of light from the display surfacein such a way as to deflect the rays of light in the height direction ofthe display surface, and the following condition is fulfilled: D/H≦0.35,where H represents the dimension of the projection surface in thedirection corresponding to the height direction of the display surface,and D represents the maximum length, in the direction normal to theprojection surface, of the space through which the rays of light pass totravel from the display surface to the projection surface. A projectionapparatus incorporating this oblique projection optical system is slimrelative to the size of the screen.

Here, the following condition may additionally be fulfilled: 30≦β≦100,where β represents the ratio of the size of the projection surface tothe size of the display surface. The symbol β represents themagnification factor by which the image is magnified by projection. Witha magnification factor lower than 30, interference between thereflecting surfaces themselves needs to be avoided by shifting, in thedirection perpendicular to the projection surface, reflecting surfacesthat are adjacent to each other in space. On the other hand, with amagnification factor over 100, it is necessary to reduce the f-numbersto secure sufficient brightness, and thus it is difficult to obtainhigher imaging performance while shortening the optical length. Byfulfilling the condition described above, it is possible both tomaintain high imaging performance and to slim down and miniaturize aprojection apparatus incorporating this oblique projection opticalsystem.

Eighth to Tenth Embodiments

Next, the rear projection optical systems of an eighth to a tenthembodiment of the invention will be described. FIGS. 40, 44, and 48 showthe entire projection path from a panel display surface I1 to a screensurface I2 in the eighth, ninth, and tenth embodiments, respectively.FIGS. 41, 45, and 49 show, in enlarged views, the projection opticalsystem constituting the eighth, ninth, and tenth embodiments,respectively, and a principal portion of the optical path thereof. Theseoptical path diagrams show optical sections taken along the Y-Z plane ofthe Cartesian coordinate system (X, Y, and Z) described later. It is tobe understood that the rear projection optical systems of theseembodiments do not necessarily have to be designed precisely as shown intheir respective optical path diagrams, but may be designed upside down;that is, turning their construction upside down to suit actualarrangement does not affect their function in any way.

The eighth to tenth embodiments deal with rear projection opticalsystems for use in rear-projection-type image projection apparatus (rearprojectors). These rear projection optical systems are provided with aprojection optical system for projecting a magnified image of atwo-dimensional image displayed on a panel display surface I1 (the imagedisplay surface of a display panel located on the reduction side) onto ascreen surface I2. The display panel is realized with a display devicesuch as a reflective liquid crystal panel, transmissive liquid crystalpanel, or DMD (digital micromirror device). The panel display surface I1is illuminated with illumination light emitted from a lamp (not shown)and passing through an illumination optical system (not shown). As thepanel display surface I1 is illuminated, projection light emanatestherefrom, which is then led to the screen surface I2 by the projectionoptical system and other components described later. Projection of acolor image is achieved by adopting a three-panel construction in whichthree display panels are arranged and color integration is achieved bythe use of a cross dichroic prism or the like, a single-panelconstruction in which an image is displayed on a time division basis, ora single-panel construction in which a microlens array is arranged on adisplay panel.

The rear projection optical systems of the eighth to tenth embodimentsinclude, from the panel display surface I1 side, a projection opticalsystem composed of a first to a fourth mirror M1 to M4 and an opticalpath turning mirror composed of a fifth and a sixth mirror M5 and M6. Inall these embodiments, the mirrors constituting the projection opticalsystem are all curved-surface reflecting mirrors, of which reflectingsurfaces are all free-form curved surfaces. Moreover, in all theseembodiments, the two mirrors constituting the optical path turningmirror are both flat-surface reflecting mirrors. The projection lightemanating from the panel display surface I1 is reflected by the fourcurved-surface reflecting mirrors constituting the projection opticalsystem, then has its optical path turned twice by the two flat-surfacereflecting mirrors, and then reaches the screen surface I2. The symbolST represents an aperture stop position ST, which corresponds to avirtual aperture stop plane.

In all these embodiments, projection of a color image is achieved, asdescribed above, by arranging a color integrating prism, such as a crossdichroic prism, near the screen surface I2. For example, illuminationlight is separated into R, G, and B light by an illumination opticalsystem so as to be separately shone on three display panels and thenintegrated back together by a cross dichroic prism. The cross dichroicprism may be used for both color separation and color integration. Whenthe display panel is of a reflective type, incident and reflected raysmay be separated by the use of a beam separating prism, such as apolarization beam splitter (PBS) or TIR (total internal reflection)prism. A condenser lens may be arranged near the panel display surfaceI1 to make the rear projection optical system telecentric toward thepanel display surface I1.

In a case where a display panel, such as a liquid crystal panel, thatexhibits different characteristics depending on the angle of incidencethereon is used, it is preferable to make the projection optical systemtelecentric toward the panel display surface I1. However, to obtainhigher optical performance, it is better to make it less telecentric.Thus, in a non-telecentric optical system, a condenser lens may bearranged in front of the panel display surface I1 to make the opticalsystem telecentric with respect to the panel display surface I1. In acase where a reflective display panel is used and beam splitting needsto be achieved without the use of a PBS, to permit incident andreflected rays to be separated on the basis of the difference betweentheir angles, the incident rays need to be inclined relative to thepanel display surface I1 at an angle larger than the angle determined bythe f-number in the direction in which the optical path is turned. Inthis case, it is preferable to adopt a so-called oblique telecentricconstruction (in which rays are incident obliquely on the panel displaysurface I1, at almost uniform angles of incidence over the entire areathereof). In an oblique telecentric construction, the aforementionedangle characteristics of liquid crystal do not matter. An obliquetelecentric construction may be realized by arranging a condenser lens(decentered as required) in front of the panel display surface I1.

An oblique projection optical system can be of one of the following sixtypes:

(i) a transmissive optical system employing part of a centered opticalsystem;

(ii) a non-axis-symmetric transmissive optical system employing a relay;

(iii) a non-axis-symmetric transmissive optical system employing norelay;

(iv) a reflective optical system employing part of a centered opticalsystem;

(v) a non-axis-symmetric reflective optical system; and

(vi) a non-axis-symmetric optical system partly reflective and partlytransmissive.

With the type (i), to obtain a large oblique projection angle as in theeighth to tenth embodiments, the original centered optical system needsto have a very wide angle of view. In general, attempting to obtainsatisfactory optical performance with a wide-angle lens results in usingmany lenses and thus in high cost. With the type (ii), i.e. anon-axis-symmetric optical system that employs a relay to eliminatetrapezoid distortion, it is necessary to form an intermediary image.This makes the projection optical system very large. With the types(iii) and (vi), oblique projection is achieved by the use of, forexample, free-form curved surfaces or the like. However, sincetransmissive optical components cause dispersion and thus chromaticaberration, it is necessary to use additional optical components tocorrect it. Thus, even more components need to be used than with thereflective types (iv) and (v). With the type (iv), no chromaticaberration appears, but, as with the type (i), the centered opticalsystem requires a very large number of lenses.

With the type (v), the reflective optical system does not causechromatic aberration. Moreover, by using free-form curved surfaces thatare decentered relative to each other, it is possible to obtainsatisfactory optical performance and distortion-free images, whichcannot be achieved with a centered optical system. To achieve this, asin the eighth to tenth embodiments, it is preferable that the projectionoptical system have at least three curved-surface reflecting mirrors,and, for maximum compactness, it is further preferable that theprojection optical system form no intermediary image in the optical pathfrom the panel display surface I1 to the screen surface I2. In general,in a reflective optical system, it is necessary to use at least onepositive and one negative reflecting mirror to correct for the Petzvalsum, and it is necessary to use another free-form curved-surface mirrorto correct for distortion; that is, using at least three reflectingmirrors in total makes it possible to realize a projection opticalsystem that offers satisfactory optical performance and that producesalmost distortion-free images.

When the projection optical system is composed of three reflectingmirrors, i.e. the least required as described above, it may be possibleto obtain satisfactory optical performance, but the projection opticalsystem is then extremely sensitive to errors inevitable in the assemblyprocess when it is manufactured. That is, its optical performancedeteriorates greatly by going through the assembly process. To avoidthis, it is preferable that, as in the eighth to tenth embodiments, theprojection optical system have at least four curved-surface reflectingmirrors. Using at least four curved-surface reflecting mirrors helpsalleviate the responsibility of each reflecting surface for thecorrection of aberrations, and helps disperse the sensitivity toassembly errors. Thus, as compared with a projection optical systemcomposed of the least required number of reflecting mirrors, it ispossible to reduce assembly errors

By giving at least three of the curved-surface reflecting mirrors afree-form curved surface, it is possible to obtain better opticalperformance. Therefore, in a reflective optical system, like those ofthe eighth to tenth embodiments, that has four or more curved-surfacereflecting mirrors, it is preferable that at least three of thosecurved-surface reflecting mirrors have a free-form curved surface. Here,a free-form curved surface denotes a surface that includes a greatlydecentered aspherical surface and that does not have an axis of rotationsymmetry near the center of its effective area, that is, a surface thatis not spherical but has aspherical undulations (freedom). Theaspherical undulations of a free-form curved surface can be exploited tocontrol the curvature of a reflecting surface three-dimensionally. Thispermits non-axis-symmetric aberrations (distortions and the like)resulting from oblique projection to be corrected for easily withsurface inclinations that are so set as to vary from point to point onthe reflecting surface.

It is further preferable that the free-form curved surfaces used as thereflecting surfaces of the curved-surface reflecting mirrors have noaxis of rotation symmetry but one plane of symmetry. In the eighth totenth embodiments, the Y-Z plane (the plane parallel to the plane oftheir respective optical path diagrams) is the plane of symmetry of eachfree-form curved surface. That is, the reflecting surface of eachcurved-surface reflecting mirror is a free-form curved surface that issymmetric about that plane of symmetry. Free-form curved surfaces likethese that are symmetric about a plane are easier to produce andevaluate than those which are not symmetric about a plane.

Moreover, in the eighth to tenth embodiments, the rear projectionoptical system as a whole is symmetric about the plane (i.e. the Y-Zplane) that runs vertically through the center of the screen. This makesthe production of optical components easy, and helps alleviate unevenbrightness and uneven distortion between the right and left parts of thescreen. However, where compactness matters as in a rear projectiontelevision set, the rear projection optical system as a whole can bemade smaller by turning the optical path in the width direction (i.e.the X direction). For example, by arranging, between the curved-surfacereflecting mirror that serves as the last surface of the projectionoptical system and the flat-surface reflecting mirror that is the firstto reflect the light exiting from the projection optical system, aflat-surface reflecting mirror that turns the optical path in the widthdirection (the X direction), it is possible to reduce the protrusion inan upper or lower portion of the rear projection television set (i.e.where the display panel and the projection optical system are arranged).Here, the optical path is turned with a flat-surface reflecting mirror,and therefore optical symmetry is not affected.

In the eighth to tenth embodiments, the optical path is turned only inthe direction perpendicular to the direction of the longer sides (i.e.the X direction) of the screen surface I2, that is, in the directionparallel to the Y-Z plane. In optical arrangements, like those of theseembodiments, where the optical path is turned with a plurality ofreflecting mirrors, the optical path needs to be turned so that rays donot overlap. However, from the viewpoint of optical performance, it isnot preferable to secure a large margin for the turning of the opticalpath, because this increases the degree of non-axis symmetry. It ispossible to obtain satisfactory optical performance by reducing thecross sectional area of the beam in the direction in which the opticalpath is turned. The cross sectional area of the beam in the direction inwhich the optical path is turned is reduced, preferably, by making theaperture stop ST elliptic. That is, the diameter of the aperture stop inthe direction in which the optical path is turned (i.e. in the directionparallel to the Y-Z plane) is reduced, and the diameter of the aperturestop in the direction perpendicular thereto (i.e. in the X direction) isincreased. Using an elliptic aperture stop like this makes it possibleto realize a rear projection optical system that offers satisfactoryoptical performance (i.e. in which the turning of the optical pathcauses little non-axis symmetry) without changing the total area (andthus the brightness) of the aperture stop.

Assume that the ray traveling from the center of the panel displaysurface I1 through the center of the aperture stop ST to the center ofthe screen surface I2 is called the “screen center ray.” Then, it ispreferable that conditional formulae (1) and (2) below be fulfilled. Ina rear projection optical system, like those of the eighth to tenthembodiments, that includes a projection optical system having at leastfour curved-surface reflecting mirrors, designing the optical system tofulfill conditional formulae (1) and (2) makes it possible to realize arear projection optical system that offers satisfactory opticalperformance with little distortion but is nevertheless advantageous interms of mass production and cost and that is slim and is composed ofcompact optical components such as reflecting mirrors.

0.5<DL/HL<3.5  (1)

10°<θ<70°  (2)

where

DL represents the optical distance traveled by the screen center rayfrom the last surface of the projection optical system to the screensurface I2;

HL represents the dimension of the screen surface I2 in the directionparallel to the plane (corresponding to the Y-Z plane in the opticalpath diagrams) formed at the center of the screen surface I2 by a normalto the screen surface I2 and the screen center ray (that is, thisdimension corresponds to the length of the shorter sides of the screensurface I2 in the eighth to tenth embodiments); and

θ represents the angle of incidence at which the screen center ray isincident on the screen surface I2.

Conditional formula (1) defines the preferable angle of view as theratio of the object distance (i.e. the projection distance) DL to thesize of the screen surface I2. If the lower limit of conditional formula(1) is transgressed, a wide angle of view is required, and therefore itis difficult to obtain satisfactory optical performance. To obtainsatisfactory optical performance, it is necessary to make the projectionoptical system as a whole longer and use larger reflecting mirrors, orto increase the number of reflective optical components used. However,either remedy leads to higher cost and is thus undesirable. Moreover, inoptical arrangements, like those of the eighth to tenth embodiments,that employ two flat-surface reflecting mirrors, it is essential tosecure a certain object distance DL to the screen surface I2 to permitthe optical path to be turned in a compact form. If conditional formula(1) is fulfilled, the optical path can be turned with the flat-surfacereflecting mirror arranged on the screen surface I2 side of theprojection optical system. This helps make the optical system as awhole, including the screen surface I2, slim and compact without undulyincreasing cost. It is preferable to fulfill conditional formula (1)with its lower limit raised to 2.5. This makes it possible to realize aslim projection optical system that offers better optical performanceand that employs inexpensive optical components.

If the upper limit of conditional formula (1) is transgressed, the angleof view is narrow. This is advantageous from the viewpoint of opticalperformance, but makes the object distance DL to the screen surface I2unnecessarily long, making the rear projection optical system as a wholelarge. It is preferable to fulfill conditional formula (1) with itsupper limit lowered to 3.2. This makes it possible to realize a morecompact rear projection optical system.

Conditional formula (2) defines the preferable oblique projection angle.If the upper limit of conditional formula (2) is transgressed, theoblique projection angle is very large. A large oblique projection anglemakes it difficult to obtain satisfactory optical performance. It ispreferable to fulfill conditional formula (2) with its upper limitlowered to 63. This makes it possible to obtain better opticalperformance.

If the lower limit of conditional formula (2) is transgressed, it iseasy to obtain satisfactory optical performance. However, rays are thenincident on the screen surface I2 from a direction close toperpendicular thereto. This makes it difficult to achieve slimming-downthrough oblique projection. It is preferable to fulfill conditionalformula (2) with its lower limit raised to 30. This makes it possible torealize a slimmer rear projection optical system. It is furtherpreferable to fulfill conditional formula (2) with its lower limitraised to 40. This makes it possible to achieve further slimming-down.It is even further preferable to fulfill conditional formula (2) withits lower limit raised to 45. This makes it possible to achieve evenfurther slimming-down.

Where, as in the eighth to tenth embodiments, an oblique projectionoptical system is used, it is preferable to use a screen suitable forthe particular rear projection optical system used to realize it.Typically, with a rear projection television set or the like is used ascreen having a Fresnel lens, a lenticular plate, and a black matrixarranged in this order from the incident side. In oblique projection, aray is incident at an angle on the center of the screen surface I2, andtherefore, as shown in FIG. 52, it is preferable either to use adecentered Fresnel lens FL having a flat-surface portion FA on its sideon which the projection light is incident (in the figure, the elementsother than the Fresnel lens FL are omitted), or to use a screen composedof a total reflection prism array and an ordinary Fresnel lens combinedtogether. If, contrary to the arrangement shown in FIG. 52, a Fresnelportion (FB) is located on the side on which the projection light isincident, vignetting occurs.

With the arrangement shown in FIG. 52, rays appear discontinuous atintervals equal to the pitch of the Fresnel lens. To alleviate thiseffect, it is preferable to make the pitch of the Fresnel lenssufficiently finer than the size of the pixels displayed on the screensurface I2. Specifically, it is preferable that the following conditionbe fulfilled: [the pitch of the Fresnel lens]/[the size of the pixels onthe screen]<0.5. It is further preferable that the following conditionbe fulfilled: [the pitch of the Fresnel lens]/[the size of the pixels onthe screen]<0.3. In the eighth to tenth embodiments, the problemdescribed above can be overcome by using, for example, a screen with apixel size of about 1 mm and a Fresnel lens pitch of about 0.2 mm.

Next, the relationship between the material of the mirrors and thetemperature characteristics of the rear projection optical system.Usually, a projector has a heat generating member in the form of a lightsource, and the individual optical components not only transmit orreflect light but also absorb a slight amount of light. Therefore, afterthe lamp is turned on, the temperature of those optical componentsrises. Moreover, the ambient temperature is never constant. Thus, it isdesired that a rear projection optical system offer satisfactory opticalperformance stably against variations in temperature. Moreover, it isgenerally known that the sensitivity to errors of reflective opticalcomponents such as reflecting mirrors is more than twice as high as thatof ordinary transmissive optical components. Therefore, in the eighth totenth embodiments, it is preferable that the curved-surface reflectingmirrors have their substrate made of glass, which exhibits relativelysmall variations in properties against variations in temperature. It ispreferable that the substrate is coated with a reflective coating suchas an enhanced reflective coating formed by vapor-depositing aluminum orsilver, or a reflective coating formed of dielectric multilayer film.However, aluminum and silver absorb a slight amount of light and thuspose a risk of generating heat. Thus, from the viewpoint of minimizingheat generation, it is preferable to use a reflective coating formed ofdielectric multilayer film.

The curved-surface reflecting mirror closest to the screen surface I2has a relatively weak optical power and thus has low sensitivity toerrors. Therefore, this curved-surface reflecting mirror may have itssubstrate made of plastic, such as PMMA (polymethyl methacrylate, PC(polycarbonate), or polyolefin resin. That is, a reflecting mirror ofwhich the substrate is made of plastic and coated with an enhancedreflective coating formed by vapor-depositing aluminum or silver may beused as the curved-surface reflecting mirror closest to the screensurface I2, because this has little effect on the optical performanceobtained. Moreover, using a plastic substrate instead of a glasssubstrate helps reduce cost. Considering the relationship describedabove between the material of the mirrors and the temperaturecharacteristics of the rear projection optical system, it is preferablethat at least the first and second curved-surface reflecting mirrors ascounted from the panel display surface I1 side have their substrate madeof glass.

Next, how focusing and zooming are achieved in the rear projectionoptical system will be described. It is preferable to achieve focusingby moving the display panel along the screen center ray, or by movingthe first or second mirror M1 or M2 along it. It is preferable toachieve zooming by moving at least two curved-surface reflectingmirrors. It is to be noted that, in a rear projection television set,the screen surface I2 is kept in a fixed position, and therefore, toadapt the display area to the screen surface I2, it is necessary toadjust the display area within a margin of a few percent by zooming.

EXAMPLES 8 TO 10

Practical examples (Examples 8 to 10) of the eighth to tenth embodimentswill be presented in detail below with reference to their constructiondata, spot diagrams, and other data. Examples 8 to 10 presented belowcorrespond to the eighth to tenth embodiments, respectively, andtherefore the figures showing those embodiments also show theconstruction and optical path of Examples 8 to 10.

Tables 59 to 64, 65 to 70, and 71 to 76 show the construction data ofExamples 8 to 10, respectively. Of these tables, Tables 59, 65, and 71show the size (mm) of the panel display surface I1, the size (mm) of thescreen surface I2, and the f-numbers (FNO) in the directions of thelonger and shorter sides of the screen (the X and Y directions,respectively). Tables 60, 66, and 72 show the data of the individualsurfaces of the respective rear projection optical systems, in orderfrom the reduction side. Tables 61, 67, and 73 show the free-form curvedsurface data representing the shape of the curved surface of the firstmirror (M1). Tables 62, 68, and 74 show the free-form curved surfacedata representing the shape of the curved surface of the second mirror(M2). Tables 63, 69, and 75 show the free-form curved surface datarepresenting the shape of the curved surface of the third mirror (M3).Tables 64, 70, and 76 show the free-form curved surface datarepresenting the shape of the curved surface of the fourth mirror (M4).The data of each surface is given in coordinates (X, Y, and Z) in aright-handed Cartesian coordinate system. Specifically, the position (X,Y, and Z coordinates) of a surfaces is given as the coordinates (mm) ofits vertex in the coordinate of which the origin (0, 0, 0) is located atthe center of the screen surface I2, and the inclination (X, Y, and Zrotation) of the surface is given as the rotation angles (°) about theX, Y, and Z axes with respect to its vertex. In the optical pathdiagrams, the X axis runs vertically to the plane of the diagrams (thedirection pointing from the front to back side of the plane of thediagrams as seen from the viewer is the positive direction, and thecounter-clockwise rotation on the plane of the diagrams as seen from theviewer is the positive X rotation). The Y axis runs along theintersection line between the screen surface I2 and the plane of thediagrams (the upward direction in the diagrams is the positivedirection), and the Z axis runs along the normal to the screen surfaceI2 (the rightward direction in the diagrams is the positive direction).

The reflecting surface of each curved-surface reflecting mirror is afree-form curved surface, of which the surface shape is defined byformula (FS) below using coordinates (x, y, and z) in the Cartesiancoordinate system having its origin at the vertex of the surface. Table77 lists the values of the conditional formulae and the related data asobserved in each example. $\begin{matrix}{z = {{\left( {c \cdot h^{2}} \right)/\left\lbrack {1 + \sqrt{1 - {\left( {1 + K} \right) \cdot c^{2} \cdot h^{2}}}} \right\rbrack} + {\sum\limits_{m}{\sum\limits_{n}\left\lbrack {{C\left( {m,n} \right)} \cdot x^{m} \cdot y^{n}} \right\rbrack}}}} & ({FS})\end{matrix}$

where

z represents the displacement from the reference surface along theoptical axis at the height of h;

h represents the height in the direction perpendicular to the opticalaxis (h²=x²+y²)

c represents the paraxial curvature (=the reciprocal of the radius ofcurvature);

K represents the conic constant, and

C(m, n) represents the free-form surface coefficients (those which equalzero are omitted).

The optical performance of Examples 8 to 10 is shown in spot diagrams inFIGS. 42, 46, and 50 and distortion diagrams in FIGS. 43, 47, and 51,respectively. The spot diagrams show the imaging characteristics (mm)observed on the screen surface I2 for light having a wavelength of 550(nm), and the distortion diagrams show the positions (mm), observed onthe screen surface I2, of the rays corresponding to a rectangular gridpattern displayed on the panel display surface I1. In the distortiondiagrams, D1 (solid lines) indicates the distortion grid of eachexample, and D0 (broken lines) indicates the grid of ideal image points(without distortion) calculated in consideration of the anamorphicratio. The object height (mm) corresponding to each field position isgiven in coordinates (x, y) in the coordinate system of which the originis located at the center of the panel display surface I1, of which the xaxis runs in the same direction as the X axis, and of which the y axisruns perpendicularly to the x axis and parallel to the panel displaysurface I1. On the other hand, the image height (mm) corresponding toeach field position is given in coordinates (x′, y′) in the coordinatesystem of which the origin is located at the center of the screensurface I2, of which the x′ axis runs in the same direction as the Xaxis, and of which the y′ axis runs perpendicularly to the x′ axis andparallel to the screen surface I2. Thus, the distortion diagrams showthe distortion (though only in the x′ negative direction) of the imageactually projected on the screen surface I2 as observed from thedirection perpendicular to the x′-y′ plane.

TABLE 1 Examples 1 to 7  Overall (1) Angle of Incidence on ProjectionSize (Half) Surface Magnifi- Magnifi- Display Surface Projection SurfaceCenter cation β cation β Anamo S0 S6 Bottom Diagonal Angle EntrancePupil Example (Y) (Z) (Y/Z) Height Width Height Width (Minimum)(Maximum) Center of View X Y 1 78.57 −78.56 −0.02% 4.36 7.75 342.4 608.824.0 67.3 52.1 43.3 ∞ −9.9° 2 71.41 −71.39 −0.02% 4.36 7.75 311.3 553.524.7 65.8 50.9 41.1 ∞ 0.0° 3 71.40 −71.39 −0.02% 4.36 7.75 311.3 553.524.3 65.7 50.7 41.4 ∞ −11.3° 4 51.23 −51.27 0.08% 4.86 8.63 249.1 442.835.9 71.4 61.3 35.5  40 −8 5 51.25 −51.27 0.03% 4.86 8.63 249.1 442.834.8 69.7 59.8 34.9  80 −20 6 70.45 −70.50 0.08% 4.86 8.63 342.4 608.840.2 73.2 64.3 33.0 1400 −250 7 51.24 −51.27 0.06% 4.86 8.63 249.1 442.836.8 69.1 58.6 32.2 ∞ 0.0°

TABLE 2 Examples 1 to 7  Overall (2) Optical Projection Path SurfaceReflecting Reflecting Reflecting Reflecting Example Fnoy Fnoz ThicknessD Height H D/H Surface S1 Surface S2 Surface S3 Surface S4 1 3.5 3.4202.4 684.9 0.30 sp(+) asp(−) exasp(+) exasp(−) 1-b 3.5 3.4 227.4 684.90.33 sp(+) asp(−) exasp(+) exasp(−) 1-c 3.5 3.4 402.4 684.9 0.59 sp(+)asp(−) exasp(+) exasp(−) 2 2.5 2.5 200.0 622.6 0.32 asp(+) asp(−)exasp(+) exasp(−) 3 2.6 2.5 200.0 622.6 0.32 asp(+) asp(−) exasp(+)exasp(−) 4 3.6 3.5 131.3 498.1 0.26 exasp(+) exasp(−) exasp(+) exasp(−)5 3.7 3.5 120.0 498.1 0.24 exasp(+) exasp(−) exasp(+) exasp(−) 6 3.4 3.3145.0 684.9 0.21 exasp(+) exasp(−) exasp(+) exasp(−) 7 2.5 2.5 152.3498.1 0.31 exasp(+) exasp(−) exasp(+) exasp(−)

TABLE 3 Example 1  Display Surface S0  N0 = N1 = 1 Local Coordinates x yz Position 0.0000 0.0000 0.0000 Vector VX 1.0000 0.0000 0.0000 VY 0.00001.0000 0.0000 VZ 0.0000 0.0000 1.0000 C0 0.000000

TABLE 4 Example 1  Reflecting Surface S1  N0 = N1 = 1 Local Coordinatesx y z Position 85.2186 0.4793 0.0000 Vector VX 0.9965 0.0838 0.0000 VY−0.0838 0.9965 0.0000 VZ 0.0000 0.0000 1.0000 C0 −0.008558

TABLE 5 Example 1  Pupil Plane APR  N0 = N1 = 1 Local Coordinates x y zPosition 28.0000 −19.5000 0.0000 Vector VX −1.0000 0.0000 0.0000 VY0.0000 1.0000 0.0000 VZ 0.0000 0.0000 −1.0000 C0 0.000000 R 8.900000

TABLE 6 Example 1  Reflecting Surface S2  N0 = N1 = 1 Local Coordinatesx y z Position −30.7050 −9.7728 0.0000 Vector VX −0.9996 0.0271 0.0000VY 0.0271 0.9996 0.0000 VZ 0.0000 0.0000 −1.0000 C0 0.007362 ε A4 A6 A8A10 A12 1.0 6.16741 × 10⁻⁷ 5.21815 × 10⁻¹⁰ −1.02686 × 10⁻¹² 2.00142 ×10⁻¹⁵ −1.17276 × 10⁻¹⁸

TABLE 7 Example 1  Reflecting Surface S3  N0 = N1 = 1 Local Coordinatesx y z Position 84.4174 3.7185 0.0000 Vector VX 0.9957 0.0926 0.0000 VY−0.0926 0.9951 0.0000 VZ 0.0000 0.0000 1.0000 C0 0.007444 ε A4 A6 A8 A10A12 1.0   7.06885 × 10⁻⁸ −2.37725 × 10⁻¹¹ −3.26077 × 10⁻¹⁵   1.19913 ×10⁻¹⁹   0.00000 × 10⁰ Bjk j = 0 j = 1 j = 2 j = 3 j = 4 k = 0   0.00000× 10⁰   0.00000 × 10⁰ −7.41333 × 10⁻³ −4.06175 × 10⁻⁵ −3.54978 × 10⁻⁷ k= 2 −5.07015 × 10⁻³ −8.65237 × 10⁻⁶ −8.90477 × 10⁻⁷ −1.88595 × 10⁻⁸−2.85271 × 10⁻¹⁰ k = 4   1.57125 × 10⁻⁷ −2.08221 × 10⁻⁹ −6.76821 × 10⁻¹⁰−2.41217 × 10⁻¹¹ −3.12921 × 10⁻¹³ k = 6 −3.66527 × 10⁻¹⁰ −2.29003 ×10⁻¹¹ −4.84278 × 10⁻¹³ −4.74385 × 10⁻¹⁵ −1.79513 × 10⁻¹⁷ k = 8   3.87171× 10⁻¹⁵   8.96657 × 10⁻¹⁷ −2.99122 × 10⁻¹⁹ k = 10   4.72013 × 10⁻¹⁹ Bjkj = 5 j = 6 j = 7 j = 8 j = 9 k = 0   2.15348 × 10⁻⁹   2.06937 × 10⁻¹¹−1.11585 × 10⁻¹² −1.32674 × 10⁻¹⁴ −1.00000 × 10⁻¹⁶ k = 2 −5.41109 ×10⁻¹² −4.87903 × 10⁻¹⁴ −4.14794 × 10⁻¹⁶ −1.81052 × 10⁻¹⁸ k = 4 −2.11459× 10⁻¹⁵ −6.34071 × 10⁻¹⁸ k = 0, j = 10 −3.57818 × 10⁻¹⁹

TABLE 8 Example 1  Reflecting Surface S4  N0 = N1 = 1 Local Coordinatesx y z Position −74.6145 15.5740 0.0000 Vector VX −0.9322 0.3619 0.0000VY 0.3619 0.9322 0.0000 VZ 0.0000 0.0000 −1.0000 C0 −0.005995 ε A4 A6 A8A10 A12 1.0 −1.69975 × 10⁻⁶   2.14404 × 10⁻¹⁰ −1.29314 × 10⁻¹⁵ −1.08992× 10⁻¹⁹   3.57834 × 10⁻²⁴ Bjk j = 0 j = 1 j = 2 j = 3 j = 4 k = 0  0.00000 × 10⁰   0.00000 × 10⁰ −4.35314 × 10⁻² −1.23693 × 10⁻³ −9.02155× 10⁻⁶ k = 2   6.98217 × 10⁻³ −4.70094 × 10⁻⁴ −7.53704 × 10⁻⁶ −4.73627 ×10⁻⁸   9.21703 × 10⁻¹³ k = 4 −2.22242 × 10⁻⁶ −3.08825 × 10⁻⁸   5.63301 ×10⁻¹⁰   1.42978 × 10⁻¹¹ −8.61211 × 10⁻¹⁴ k = 6   1.35876 × 10⁻⁹  4.88064 × 10⁻¹¹   5.67002 × 10⁻¹³   2.57805 × 10⁻¹⁵   3.29139 × 10⁻¹⁸k = 8 −7.02524 × 10⁻¹⁴ −1.36769 × 10⁻¹⁵ −7.18941 × 10⁻¹⁸ k = 10  9.65035 × 10⁻¹⁹ Bjk j = 5 j = 6 j = 7 j = 8 j = 9 k = 0   4.57518 ×10⁻⁸   1.44673 × 10⁻⁹   1.23054 × 10⁻¹¹   2.13590 × 10⁻¹⁴ −1.82326 ×10⁻¹⁶ k = 2   1.20840 × 10⁻¹² −1.15121 × 10⁻¹³ −1.24718 × 10⁻¹⁵ −3.74717× 10⁻¹⁸ k = 4 −2.11513 × 10⁻¹⁵ −8.26515 × 10⁻¹⁸ k = 0, j = 10 −7.11582 ×10⁻¹⁹

TABLE 9 Example 1  Reflecting Surface S5  N0 = N1 = 1 Local Coordinatesx y z Position 90.6219 0.0000 0.0000 Vector VX 1.0000 0.0000 0.0000 VY0.0000 1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C0 0.000000

TABLE 10 Example 1  Projection Surface S6  N0 = N1 = 1 Local Coordinatesx y z Position −109.3781 −547.8375 0.0000 Vector VX −1.0000 0.00000.0000 VY 0.0000 −1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C0 0.000000

TABLE 11 Example 2  Display Surface S0  N0 = N1 = 1 Local Coordinates xy z Position 0.0000 0.0000 0.0000 Vector VX 1.0000 0.0000 0.0000 VY0.0000 1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C0 0.000000

TABLE 12 Example 2  Reflecting Surface S1  N0 = N1 = 1 Local Coordinatesx y z Position 115.4561 −17.9518 0.0000 Vector VX 1.0000 0.0024 0.0000VY −0.0024 1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C0 −0.007256 ε A4 A6 A8A10 A12 1.0 −3.03358 × 10⁻⁹ −1.39052 × 10⁻¹² 6.22246 × 10⁻¹⁶ −1.50263 ×10⁻¹⁹ 0.00000 × 10⁰

TABLE 13 Example 2  Pupil Plane APR  N0 = N1 = 1 Local Coordinates x y zPosition 48.0000 −17.7500 0.0000 Vector VX −1.0000 0.0000 0.0000 VY0.0000 1.0000 0.0000 VZ 0.0000 0.0000 −1.0000 C0  0.000000 R 14.500000

TABLE 14 Example 2  Reflecting Surface S2  N0 = N1 = 1 Local Coordinatesx y z Position 4.7679 −14.4679 0.0000 Vector VX −0.9998 −0.0206 0.0000VY −0.0206 0.9998 0.0000 VZ 0.0000 0.0000 −1.0000 C0 0.006847 ε A4 A6 A8A10 A12 1.0 4.61969 × 10⁻⁷ −5.13932 × 10⁻¹⁰ 1.06677 × 10⁻¹² −1.04751 ×10⁻¹⁵ 4.52949 × 10⁻¹⁹

TABLE 15 Example 2  Reflecting Surface S3  N0 = N1 = 1 Local Coordinatesx y z Position 112.9764 −7.6544 0.0000 Vector VX 0.9995 −0.0313 0.0000VY 0.0313 0.9995 0.0000 VZ 0.0000 0.0000 1.0000 C0 0.007888 ε A4 A6 A8A10 A12 1.0   3.40076 × 10⁻⁸ −2.52102 × 10⁻¹¹ −3.48433 × 10⁻¹⁵   1.13295× 10⁻¹⁹   0.00000 × 10⁰ Bjk j = 0 j = 1 j = 2 j = 3 j = 4 k = 0  0.00000 × 10⁰   0.00000 × 10⁰ −6.77427 × 10⁻³ −4.68604 × 10⁻⁵ −4.89629× 10⁻⁷ k = 2 −2.73509 × 10⁻³   5.65785 × 10⁻⁵ −3.30783 × 10⁻⁷ −2.18317 ×10⁻⁸ −3.01333 × 10⁻¹⁰ k = 4 −6.35040 × 10⁻⁷ −3.94313 × 10⁻⁸ −1.00084 ×10⁻⁹ −2.18172 × 10⁻¹¹ −3.02263 × 10⁻¹³ k = 6 −6.48372 × 10⁻¹⁰ −3.60308 ×10⁻¹¹ −6.03649 × 10⁻¹³ −4.89780 × 10⁻¹⁵ −1.65871 × 10⁻¹⁷ k = 8 −7.06784× 10⁻¹⁴   2.28215 × 10⁻¹⁵   1.50126 × 10⁻¹⁷ k = 10   5.38948 × 10⁻¹⁷ Bjkj = 5 j = 6 j = 7 j = 8 j = 9 k = 0   1.29013 × 10⁻⁹   2.07728 × 10⁻¹¹−1.18182 × 10⁻¹² −1.36299 × 10⁻¹⁴ −1.01150 × 10⁻¹⁶ k = 2 −5.30638 ×10⁻¹² −5.02627 × 10⁻¹⁴ −4.50028 × 10⁻¹⁶ −1.91401 × 10⁻¹⁸ k = 4 −2.62883× 10⁻¹⁵ −9.73344 × 10⁻¹⁸ k = 0, j = 10 −3.57759 × 10⁻¹⁹

TABLE 16 Example 2  Reflecting Surface S4  N0 = N1 = 1 Local Coordinatesx y z Position −36.5184 −5.1790 0.0000 Vector VX −0.9261 0.3774 0.0000VY 0.3774 0.9261 0.0000 VZ 0.0000 0.0000 −1.0000 C0 −0.005895 ε A4 A6 A8A10 A12 1.0 −1.69630 × 10⁻⁶   2.14389 × 10⁻¹⁰ −1.30414 × 10⁻¹⁵ −1.08272× 10⁻¹⁹   3.51073 × 10⁻²⁴ Bjk j = 0 j = 1 j = 2 j = 3 j = 4 k = 0  0.00000 × 10⁰   0.00000 × 10⁰ −4.33336 × 10⁻² −1.23194 × 10⁻³ −9.02325× 10⁻⁶ k = 2   6.90051 × 10⁻³ −4.56827 × 10⁻⁴ −7.48434 × 10⁻⁶ −4.77783 ×10⁻⁸   1.08881 × 10⁻¹² k = 4 −2.44526 × 10⁻⁶ −3.52616 × 10⁻⁸   5.55124 ×10⁻¹⁰   1.44621 × 10⁻¹¹ −8.61666 × 10⁻¹⁴ k = 6   1.37373 × 10⁻⁹  4.93825 × 10⁻¹¹   5.65623 × 10⁻¹³   2.51280 × 10⁻¹⁵   3.10833 × 10⁻¹⁸k = 8 −6.13527 × 10⁻¹⁴ −1.31218 × 10⁻¹⁵ −7.26368 × 10⁻¹⁸ k = 10  7.45150 × 10⁻¹⁹ Bjk j = 5 j = 6 j = 7 j = 8 j = 9 k = 0   4.56377 ×10⁻⁸   1.44658 × 10⁻⁹   1.23067 × 10⁻¹¹   2.13398 × 10⁻¹⁴ −1.82367 ×10⁻¹⁶ k = 2   1.23589 × 10⁻¹² −1.15271 × 10⁻¹³ −1.24738 × 10⁻¹⁵ −3.73274× 10⁻¹⁸ k = 4 −2.11987 × 10⁻¹⁵ −8.25915 × 10⁻¹⁸ k = 0, j = 10 −7.10118 ×10⁻¹⁹

TABLE 17 Example 2  Reflecting Surface S5  N0 = N1 = 1 Local Coordinatesx y z Position 124.2382 0.0000 0.0000 Vector VX 1.0000 0.0000 0.0000 VY0.0000 1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C0 0.000000

TABLE 18 Example 1  Projection Surface S6  N0 = N1 = 1 Local Coordinatesx y z Position −75.7618 −540.0081 0.0000 Vector VX −1.0000 0.0000 0.0000VY 0.0000 −1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C0 0.000000

TABLE 19 Example 3  Display Surface S0  N0 = N1 = 1 Local Coordinates xy z Position 0.0000 0.0000 0.0000 Vector VX 1.0000 0.0000 0.0000 VY0.0000 1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C0 0.000000

TABLE 20 Example 3  Reflecting Surface S1  N0 = N1 = 1 Local Coordinatesx y z Position 101.6282 −16.6934 0.0000 Vector VX 0.9998 −0.0195 0.0000VY 0.0195 0.9998 0.0000 VZ 0.0000 0.0000 1.0000 C0 −0.007814 ε A4 A6 A8A10 A12 1.0 −4.90903 × 10⁻⁹ 1.95361 × 10⁻¹² −4.94963 × 10⁻¹⁵ 3.52855 ×10⁻¹⁸ 0.00000 × 10⁰

TABLE 21 Example 3  Pupil Plane APR  N0 = N1 = 1 Local Coordinates x y zPosition 36.0000 −26.5000 0.0000 Vector VX −1.0000 0.0000 0.0000 VY0.0000 1.0000 0.0000 VZ 0.0000 0.0000 −1.0000 C0  0.000000 R 12.800000

TABLE 22 Example 3  Reflecting Surface S2  N0 = N1 = 1 Local Coordinatesx y z Position −6.9522 −12.2530 0.0000 Vector VX −0.9999 0.0106 0.0000VY 0.0106 0.9999 0.0000 VZ 0.0000 0.0000 −1.0000 C0 0.006784 ε A4 A6 A8A10 A12 1.0 4.94051 × 10⁻⁷ −1.33387 × 10⁻¹⁰ 1.43402 × 10⁻¹³ 5.16304 ×10⁻¹⁷ −3.82708 × 10⁻²⁰

TABLE 23 Example 3  Reflecting Surface S3  N0 = N1 = 1 Local Coordinatesx y z Position 97.7147 −10.1564 0.0000 Vector VX 0.9997 0.0251 0.0000 VY−0.0251 0.9997 0.0000 VZ 0.0000 0.0000 1.0000 C0 0.008393 ε A4 A6 A8 A10A12 1.0 −3.31188 × 10⁻⁸ −3.07605 × 10⁻¹¹ −3.91714 × 10⁻¹⁵   9.45955 ×10⁻²⁰   0.00000 × 10⁰ Bjk j = 0 j = 1 j = 2 j = 3 j = 4 k = 0   0.00000× 10⁰   0.00000 × 10⁰ −7.85975 × 10⁻³ −5.59186 × 10⁻⁵ −5.77117 × 10⁻⁷ k= 2 −4.37752 × 10⁻³   2.25556 × 10⁻⁵ −4.93592 × 10⁻⁷ −2.36270 × 10⁻⁸−3.22127 × 10⁻¹⁰ k = 4 −4.43585 × 10⁻⁸ −2.03477 × 10⁻⁸ −9.41646 × 10⁻¹⁰−2.57965 × 10⁻¹¹ −3.33737 × 10⁻¹³ k = 6 −5.97881 × 10⁻¹⁰ −3.82987 ×10⁻¹¹ −7.98462 × 10⁻¹³ −7.46051 × 10⁻¹⁵ −2.63449 × 10⁻¹⁷ k = 8 −4.13400× 10⁻¹⁴ −3.34889 × 10⁻¹⁶ −1.19825 × 10⁻¹⁸ k = 10   6.82597 × 10⁻¹⁸ Bjk j= 5 j = 6 j = 7 j = 8 j = 9 k = 0 −5.12096 × 10⁻¹⁰   1.34529 × 10⁻¹¹−1.26912 × 10⁻¹² −1.42047 × 10⁻¹⁴ −1.08131 × 10⁻¹⁶ k = 2 −5.54710 ×10⁻¹² −5.20931 × 10⁻¹⁴ −5.25768 × 10⁻¹⁶ −2.30273 × 10⁻¹⁸ k = 4 −2.58191× 10⁻¹⁵ −8.83935 × 10⁻¹⁸ k = 0, j = 10 −3.70085 × 10⁻¹⁹

TABLE 24 Example 3  Reflecting Surface S4  N0 = N1 = 1 Local Coordinatesx y z Position −49.4975 −0.7471 0.0000 Vector VX −0.9339 0.3576 0.0000VY 0.3576 0.9339 0.0000 VZ 0.0000 0.0000 −1.0000 C0 −0.005880 ε A4 A6 A8A10 A12 1.0 −1.69296 × 10⁻⁶   2.14551 × 10⁻¹⁰ −1.31939 × 10⁻¹⁵ −1.07605× 10⁻¹⁹   3.47888 × 10⁻²⁴ Bjk j = 0 j = 1 j = 2 j = 3 j = 4 k = 0  0.00000 × 10⁰   0.00000 × 10⁰ −4.28053 × 10⁻² −1.22611 × 10⁻³ −9.01279× 10⁻⁶ k = 2   6.94508 × 10⁻³ −4.59022 × 10⁻⁴ −7.53319 × 10⁻⁶ −4.80275 ×10⁻⁸   1.76668 × 10⁻¹² k = 4 −2.36282 × 10⁻⁶ −3.18743 × 10⁻⁸   5.97328 ×10⁻¹⁰   1.45260 × 10⁻¹¹ −8.80551 × 10⁻¹⁴ k = 6   1.38150 × 10⁻⁹  4.92942 × 10⁻¹¹   5.59369 × 10⁻¹³   2.44035 × 10⁻¹⁵   2.82812 × 10⁻¹⁸k = 8 −6.44115 × 10⁻¹⁴ −1.37089 × 10⁻¹⁵ −7.33570 × 10⁻¹⁸ k = 10  3.26076 × 10⁻¹⁹ Bjk j = 5 j = 6 j = 7 j = 8 j = 9 k = 0   4.55202 ×10⁻⁸   1.44670 × 10⁻⁹   1.23094 × 10⁻¹¹   2.13073 × 10⁻¹⁴ −1.82245 ×10⁻¹⁶ k = 2   1.24503 × 10⁻¹² −1.15306 × 10⁻¹³ −1.24783 × 10⁻¹⁵ −3.73197× 10⁻¹⁸ k = 4 −2.12332 × 10⁻¹⁵ −8.21184 × 10⁻¹⁸ k = 0, j = 10 −7.07774 ×10⁻¹⁹

TABLE 25 Example 3  Reflecting Surface S5  N0 = N1 = 1 Local Coordinatesx y z Position 111.5073 0.0000 0.0000 Vector VX 1.0000 0.0000 0.0000 VY0.0000 1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C0 0.000000

TABLE 26 Example 3  Projection Surface S6  N0 = N1 = 1 Local Coordinatesx y z Position −88.4927 −532.1060 0.0000 Vector VX −1.0000 0.0000 0.0000VY 0.0000 −1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C0 0.000000

TABLE 27 Example 4  Display Surface S0  N0 = N1 = 1 Local Coordinates xy z Position 0.0000 0.0000 0.0000 Vector VX 1.0000 0.0000 0.0000 VY0.0000 1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C0 0.000000

TABLE 28 Example 4  Pupil Plane APR  N0 = N1 = 1 Local Coordinates x y zPosition 40.0000 −8.0000 0.0000 Vector VX 1.0000 0.0000 0.0000 VY 0.00001.0000 0.0000 VZ 0.0000 0.0000 1.0000 C0 0.000000 R 6.750000

TABLE 29 Example 4  Reflecting Surface S1  N0 = N1 = 1 Local Coordinatesx y z Position 101.8029 −20.3753 0.0000 Vector VX 1.0000 0.0053 0.0000VY −0.0053 1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C0 −0.012573 ε A4 A6 A8A10 A12 1.0   3.93713 × 10⁻⁷   1.49159 × 10⁻⁹   9.71942 × 10⁻¹⁵ −4.32620× 10⁻¹⁸   0.00000 × 10⁰ Bjk j = 0 j = 1 j = 2 j = 3 j = 4 k = 0  0.00000 × 10⁰   0.00000 × 10⁰   2.00884 × 10⁻³   1.91578 × 10⁻⁶−2.40364 × 10⁻⁷ k = 2   1.59252 × 10⁻³   1.65260 × 10⁻⁶ −5.04336 × 10⁻⁷  4.76042 × 10⁻¹¹ −4.40812 × 10⁻⁹ k = 4 −2.70902 × 10⁻⁷ −3.30009 × 10⁻¹⁰−4.42708 × 10⁻⁹   0.00000 × 10⁰   0.00000 × 10⁰ k = 6 −1.47575 × 10⁻⁹  0.00000 × 10⁰   0.00000 × 10⁰   0.00000 × 10⁰   0.00000 × 10⁰ k = 8  0.00000 × 10⁰   0.00000 × 10⁰   0.00000 × 10⁰ k = 10   0.00000 × 10⁰Bjk j = 5 j = 6 j = 7 j = 8 j = 9 k = 0   2.20863 × 10⁻¹⁰ −1.47127 ×10⁻⁹   0.00000 × 10⁰   0.00000 × 10⁰   0.00000 × 10⁰ k = 2   0.00000 ×10⁰   0.00000 × 10⁰   0.00000 × 10⁰   0.00000 × 10⁰ k = 4   0.00000 ×10⁰   0.00000 × 10⁰ k = 0, j = 10   0.00000 × 10⁰

TABLE 30 Example 4  Reflecting Surface S2  N0 = N1 = 1 Local Coordinatesx y z Position 12.5841 −39.2297 0.0000 Vector VX −0.9952 0.0978 0.0000VY 0.0978 0.9952 0.0000 VZ 0.0000 0.0000 −1.0000 C0 0.044024 ε A4 A6 A8A10 A12 1.0 −4.87501 × 10⁻⁴ −4.32896 × 10⁻⁶ −6.34246 × 10⁻¹¹   1.11218 ×10⁻¹³ −2.01234 × 10⁻¹⁶ Bjk j = 0 j = 1 j = 2 j = 3 j = 4 k = 0   0.00000× 10⁰   0.00000 × 10⁰ −1.42523 × 10⁻² −1.32737 × 10⁻⁴   4.81629 × 10⁻⁴ k= 2 −9.49763 × 10⁻³ −2.41389 × 10⁻⁴   9.64792 × 10⁻⁴ −9.61107 × 10⁻⁷  1.30127 × 10⁻⁵ k = 4   4.84935 × 10⁻⁴ −6.98502 × 10⁻⁷   1.30266 × 10⁻⁵  0.00000 × 10⁰   0.00000 × 10⁰ k = 6   4.34057 × 10⁻⁶   0.00000 × 10⁰  0.00000 × 10⁰   0.00000 × 10⁰   0.00000 × 10⁰ k = 8   0.00000 × 10⁰  0.00000 × 10⁰   0.00000 × 10⁰ k = 10   0.00000 × 10⁰ Bjk j = 5 j = 6 j= 7 j = 8 j = 9 k = 0 −3.44610 × 10⁻⁷   4.33854 × 10⁻⁶   0.00000 × 10⁰  0.00000 × 10⁰   0.00000 × 10⁰ k = 2   0.00000 × 10⁰   0.00000 × 10⁰  0.00000 × 10⁰   0.00000 × 10⁰ k = 4   0.00000 × 10⁰   0.00000 × 10⁰ k= 0, j = 10   0.00000 × 10⁰

TABLE 31 Example 4  Reflecting Surface S3  N0 = N1 = 1 Local Coordinatesx y z Position 101.8026 −77.3632 0.0000 Vector VX 0.9951 −0.0986 0.0000VY 0.0986 0.9951 0.0000 VZ 0.0000 0.0000 1.0000 C0 0.008620 ε A4 A6 A8A10 A12 1.0   2.14466 × 10⁻⁷ −4.67228 × 10⁻¹¹ −4.63566 × 10⁻¹⁵   4.77752× 10⁻¹⁸   0.00000 × 10⁰ Bjk j = 0 j = 1 j = 2 j = 3 j = 4 k = 0  0.00000 × 10⁰   0.00000 × 10⁰ −6.61267 × 10⁻³ −6.62783 × 10⁻⁶ −3.40700× 10⁻⁷ k = 2 −6.95327 × 10⁻³ −1.54197 × 10⁻⁶ −5.76526 × 10⁻⁷   5.19801 ×10⁻⁹   1.74131 × 10⁻¹⁰ k = 4 −2.81046 × 10⁻⁷   1.87703 × 10⁻⁹   1.34552× 10⁻¹⁰ −6.69182 × 10⁻¹³   2.12931 × 10⁻¹⁴ k = 6   3.99774 × 10⁻¹¹−2.38038 × 10⁻¹³   1.84027 × 10⁻¹⁴   5.42145 × 10⁻¹⁸ −4.90278 × 10⁻¹⁷ k= 8   4.38169 × 10⁻¹⁵   1.52645 × 10⁻¹⁷ −2.38690 × 10⁻¹⁷ k = 10 −4.74121× 10⁻¹⁸ Bjk j = 5 j = 6 j = 7 j = 8 j = 9 k = 0   3.39988 × 10⁻⁹  5.21016 × 10⁻¹¹ −8.25123 × 10⁻¹³ −2.45323 × 10⁻¹⁵ −2.47599 × 10⁻¹⁸ k =2 −1.62011 × 10⁻¹²   1.72914 × 10⁻¹⁴   5.21144 × 10⁻¹⁶ −2.41525 × 10⁻¹⁷k = 4   2.58344 × 10⁻¹⁶ −4.16200 × 10⁻¹⁷ k = 0, j = 10 −2.65756 × 10⁻¹⁸

TABLE 32 Example 4  Reflecting Surface S4  N0 = N1 = 1 Local Coordinatesx y z Position 6.3346 −97.3170 0.0000 Vector VX −0.9085 0.4179 0.0000 VY0.4179 0.9085 0.0000 VZ 0.0000 0.0000 −1.0000 C0 −0.001672 ε A4 A6 A8A10 A12 1.0   1.93910 × 10⁻⁷   4.40895 × 10⁻¹¹ −4.60538 × 10⁻¹⁵ −6.66318× 10⁻¹⁸ −1.04921 × 10⁻²² Bjk j = 0 j = 1 j = 2 j = 3 j = 4 k = 0  0.00000 × 10⁰   0.00000 × 10⁰   3.63408 × 10⁻³   5.60033 × 10⁻⁵  1.09617 × 10⁻⁶ k = 2   7.85586 × 10⁻³   1.30899 × 10⁻⁴   2.58152 ×10⁻⁶   5.89453 × 10⁻⁸   8.20095 × 10⁻¹⁰ k = 4 −3.89420 × 10⁻⁷ −1.63610 ×10⁻⁸ −1.04482 × 10⁻⁹ −3.48702 × 10⁻¹¹ −6.89304 × 10⁻¹³ k = 6 −4.95414 ×10⁻¹¹   3.81251 × 10⁻¹⁴   1.25618 × 10⁻¹³   6.66734 × 10⁻¹⁵   1.79935 ×10⁻¹⁶ k = 8   3.06365 × 10⁻¹⁵   1.70276 × 10⁻¹⁶   3.83915 × 10⁻¹⁷ k = 10  8.08880 × 10⁻¹⁸ Bjk j = 5 j = 6 j = 7 j = 8 j = 9 k = 0   4.09506 ×10⁻⁸   9.02998 × 10⁻¹⁰ −4.18456 × 10⁻¹² −4.82726 × 10⁻¹³ −6.37146 ×10⁻¹⁵ k = 2 −2.47675 × 10⁻¹² −7.18032 × 10⁻¹³ −1.31925 × 10⁻¹⁴   9.14530× 10⁻¹⁸ k = 4 −1.05752 × 10⁻¹⁴ −5.63936 × 10⁻¹⁷ k = 0, j = 10 −2.80979 ×10⁻¹⁷

TABLE 33 Example 4  Reflecting Surface S5  N0 = N1 = 1 Local Coordinatesx y z Position 101.8029 0.0000 0.0000 Vector VX 1.0000 0.0000 0.0000 VY0.0000 1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C0 0.000000

TABLE 34 Example 4  Projection Surface S6  N0 = N1 = 1 Local Coordinatesx y z Position −8.1971 −472.0624 0.0000 Vector VX −1.0000 0.0000 0.0000VY 0.0000 −1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C0 0.000000

TABLE 35 Example 5  Display Surface S0  N0 = N1 = 1 Local Coordinates xy z Position 0.0000 0.0000 0.0000 Vector VX 1.0000 0.0000 0.0000 VY0.0000 1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C0 0.000000

TABLE 36 Example 5  Reflecting Surface S1  N0 = N1 = 1 Local Coordinatesx y z Position 63.5704 −15.8836 0.0000 Vector VX 0.9989 −0.0460 0.0000VY 0.0460 0.9989 0.0000 VZ 0.0000 0.0000 1.0000 C0 −0.010935 ε A4 A6 A8A10 A12 1.0   1.25724 × 10⁻⁵   1.86170 × 10⁻⁷ −4.05135 × 10⁻¹²   3.01351× 10⁻¹⁴   0.00000 × 10⁰ Bjk j = 0 j = 1 j = 2 j = 3 j = 4 k = 0  0.00000 × 10⁰   0.00000 × 10⁰ −4.43584 × 10⁻⁴   2.31983 × 10⁻⁶−1.26143 × 10⁻⁵ k = 2 −1.34491 × 10⁻³   1.89361 × 10⁻⁶ −2.53874 × 10⁻⁵  1.79014 × 10⁻⁹ −5.57511 × 10⁻⁷ k = 4 −1.27249 × 10⁻⁵ −7.53181 × 10⁻¹⁰−5.57968 × 10⁻⁷   0.00000 × 10⁰   0.00000 × 10⁰ k = 6 −1.86192 × 10⁻⁷  0.00000 × 10⁰   0.00000 × 10⁰   0.00000 × 10⁰   0.00000 × 10⁰ k = 8  0.00000 × 10⁰   0.00000 × 10⁰   0.00000 × 10⁰ k = 10   0.00000 × 10⁰Bjk j = 5 j = 6 j = 7 j = 8 j = 9 k = 0   1.23098 × 10⁻⁹ −1.86105 × 10⁻⁷  0.00000 × 10⁰   0.00000 × 10⁰   0.00000 × 10⁰ k = 2   0.00000 × 10⁰  0.00000 × 10⁰   0.00000 × 10⁰   0.00000 × 10⁰ k = 4   0.00000 × 10⁰  0.00000 × 10⁰ k = 0, j = 10   0.00000 × 10⁰

TABLE 37 Example 5  Pupil Plane APR  N0 = N1 = 1 Local Coordinates x y zPosition 50.0000 −17.0000 0.0000 Vector VX −1.0000 0.0000 0.0000 VY0.0000 1.0000 0.0000 VZ 0.0000 0.0000 −1.0000 C0 0.000000 R 8.700000

TABLE 38 Example 5  Reflecting Surface S2  N0 = N1 = 1 Local Coordinatesx y z Position −1.4357 −25.9385 0.0000 Vector VX −0.9926 0.1216 0.0000VY 0.1216 0.9926 0.0000 VZ 0.0000 0.0000 −1.0000 C0 0.043194 ε A4 A6 A8A10 A12 1.0 −4.52966 × 10⁻⁴ −4.18988 × 10⁻⁶   7.40925 × 10⁻¹¹ −7.60897 ×10⁻¹³   1.94262 × 10⁻¹⁵ Bjk j = 0 j = 1 j = 2 j = 3 j = 4 k = 0  0.00000 × 10⁰   0.00000 × 10⁰ −1.66596 × 10⁻² −9.46800 × 10⁻⁵  4.45974 × 10⁻⁴ k = 2 −9.69251 × 10⁻³ −2.44519 × 10⁻⁴   8.96345 × 10⁻⁴−5.70625 × 10⁻⁷   1.25560 × 10⁻⁵ k = 4   4.55105 × 10⁻⁴ −4.81406 × 10⁻⁷  1.25666 × 10⁻⁵   0.00000 × 10⁰   0.00000 × 10⁰ k = 6   4.18834 × 10⁻⁶  0.00000 × 10⁰   0.00000 × 10⁰   0.00000 × 10⁰   0.00000 × 10⁰ k = 8  0.00000 × 10⁰   0.00000 × 10⁰   0.00000 × 10⁰ k = 10   0.00000 × 10⁰Bjk j = 5 j = 6 j = 7 j = 8 j = 9 k = 0 −1.35981 × 10⁻⁷   4.17880 × 10⁻⁶  0.00000 × 10⁰   0.00000 × 10⁰   0.00000 × 10⁰ k = 2   0.00000 × 10⁰  0.00000 × 10⁰   0.00000 × 10⁰   0.00000 × 10⁰ k = 4   0.00000 × 10⁰  0.00000 × 10⁰ k = 0, j = 10   0.00000 × 10⁰

TABLE 39 Example 5  Reflecting Surface S3  N0 = N1 = 1 Local Coordinatesx y z Position 100.3365 −68.4122 0.0000 Vector VX 0.9925 0.1224 0.0000VY −0.1224 0.9925 0.0000 VZ 0.0000 0.0000 1.0000 C0 0.006364 ε A4 A6 A8A10 A12 1.0   2.28056 × 10⁻⁷ −4.57134 × 10⁻¹¹ −4.50292 × 10⁻¹⁵   4.78237× 10⁻¹⁸   0.00000 × 10⁰ Bjk j = 0 j = 1 j = 2 j = 3 j = 4 k = 0  0.00000 × 10⁰   0.00000 × 10⁰ −3.89379 × 10⁻³ −6.30657 × 10⁻⁶ −3.77568× 10⁻⁷ k = 2 −4.45623 × 10⁻³   9.38273 × 10⁻⁶ −4.65719 × 10⁻⁷   2.03768× 10⁻⁹   1.63626 × 10⁻¹⁰ k = 4 −3.03236 × 10⁻⁷ −5.64446 × 10⁻¹⁰  1.27401 × 10⁻¹⁰ −6.31578 × 10⁻¹⁴   2.56344 × 10⁻¹⁴ k = 6   4.54551 ×10⁻¹¹ −1.90082 × 10⁻¹⁴   1.86998 × 10⁻¹⁴ −4.24892 × 10⁻¹⁸ −4.78662 ×10⁻¹⁷ k = 8   4.45436 × 10⁻¹⁵   5.48344 × 10⁻¹⁸ −2.39022 × 10⁻¹⁷ k = 10−4.77522 × 10⁻¹⁸ Bjk j = 5 j = 6 j = 7 j = 8 j = 9 k = 0   5.53503 ×10⁻¹⁰   6.99868 × 10⁻¹¹   8.59898 × 10⁻¹³   4.36970 × 10⁻¹⁴   1.30934 ×10⁻¹⁵ k = 2 −3.30206 × 10⁻¹⁴   1.85583 × 10⁻¹⁴ −2.97697 × 10⁻¹⁶ −2.94138× 10⁻¹⁷ k = 4   7.08908 × 10⁻¹⁷ −4.68875 × 10⁻¹⁷ k = 0, j = 10   1.05237× 10⁻¹⁷

TABLE 40 Example 5  Reflecting Surface S4  N0 = N1 = 1 Local Coordinatesx y z Position 19.9467 −128.3830 0.0000 Vector VX −0.9798 0.2000 0.0000VY 0.2000 0.9798 0.0000 VZ 0.0000 0.0000 −1.0000 C0 −0.004794 ε A4 A6 A8A10 A12 1.0 −1.81730 × 10⁻⁸   6.89428 × 10⁻¹¹   5.19899 × 10⁻¹⁵ −7.40999× 10⁻¹⁸ −2.99449 × 10⁻²⁵ Bjk j = 0 j = 1 j = 2 j = 3 j = 4 k = 0  0.00000 × 10⁰   0.00000 × 10⁰   3.09395 × 10⁻³   1.11583 × 10⁻⁵  1.03114 × 10⁻⁷ k = 2   5.05181 × 10⁻³   2.55058 × 10⁻⁵   4.10929 ×10⁻⁷   4.51142 × 10⁻⁹ −1.86926 × 10⁻¹⁰ k = 4 −3.04900 × 10⁻⁹ −1.36475 ×10⁻⁹ −2.42120 × 10⁻¹⁰ −8.74626 × 10⁻¹³ −3.84016 × 10⁻¹⁴ k = 6 −6.57427 ×10⁻¹¹   5.94997 × 10⁻¹⁴ −1.98078 × 10⁻¹⁴   4.55095 × 10⁻¹⁷   7.45904 ×10⁻¹⁷ k = 8 −5.36094 × 10⁻¹⁵ −2.64065 × 10⁻¹⁸   3.70927 × 10⁻¹⁷ k = 10  7.41820 × 10⁻¹⁸ Bjk j = 5 j = 6 j = 7 j = 8 j = 9 k = 0   5.43111 ×10⁻⁹   1.88009 × 10⁻¹⁰   1.91127 × 10⁻¹² −8.48082 × 10⁻¹⁴ −1.57209 ×10⁻¹⁵ k = 2 −7.19401 × 10⁻¹³ −3.38507 × 10⁻¹⁴ −7.93454 × 10⁻¹⁷   3.69191× 10⁻¹⁷ k = 4   1.73316 × 10⁻¹⁷   7.44125 × 10⁻¹⁷ k = 0, j = 10 −7.22437× 10⁻¹⁹

TABLE 41 Example 5  Reflecting Surface S5  N0 = N1 = 1 Local Coordinatesx y z Position 103.5699 0.0000 0.0000 Vector VX 1.0000 0.0000 0.0000 VY0.0000 1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C0 0.000000

TABLE 42 Example 5  Projection Surface S6  N0 = N1 = 1 Local Coordinatesx y z Position −16.4301 −478.3923 0.0000 Vector VX −1.0000 0.0000 0.0000VY 0.0000 −1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C0 0.000000

TABLE 43 Example 6  Display Surface S0  N0 = N1 = 1 Local Coordinates xy z Position 0.0000 0.0000 0.0000 Vector VX 1.0000 0.0000 0.0000 VY0.0000 1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C0 0.000000

TABLE 44 Example 6  Reflecting Surface S1  N0 = N1 = 1 Local Coordinatesx y z Position 62.4807 −11.1468 0.0000 Vector VX 0.9999 −0.0152 0.0000VY 0.0152 0.9999 0.0000 VZ 0.0000 0.0000 1.0000 C0 −0.008352 ε A4 A6 A8A10 A12 1.0   1.95222 × 10⁻⁵   1.95773 × 10⁻⁷ −3.59070 × 10⁻¹³   3.89395× 10⁻¹⁶   0.00000 × 10⁰ Bjk j = 0 j = 1 j = 2 j = 3 j = 4 k = 0  0.00000 × 10⁰   0.00000 × 10⁰ −2.00942 × 10⁻³   2.13947 × 10⁻⁶−1.97120 × 10⁻⁵ k = 2 −2.74651 × 10⁻³   1.56985 × 10⁻⁶ −3.94978 × 10⁻⁵  7.56232 × 10⁻¹⁰ −5.86884 × 10⁻⁷ k = 4 −1.97985 × 10⁻⁵ −1.18174 × 10⁻⁹−5.87120 × 10⁻⁷   0.00000 × 10⁰   0.00000 × 10⁰ k = 6 −1.95698 × 10⁻⁷  0.00000 × 10⁰   0.00000 × 10⁰   0.00000 × 10⁰   0.00000 × 10⁰ k = 8  0.00000 × 10⁰   0.00000 × 10⁰   0.00000 × 10⁰ k = 10   0.00000 × 10⁰Bjk j = 5 j = 6 j = 7 j = 8 j = 9 k = 0   1.03253 × 10⁻⁹ −1.95633 × 10⁻⁷  0.00000 × 10⁰   0.00000 × 10⁰   0.00000 × 10⁰ k = 2   0.00000 × 10⁰  0.00000 × 10⁰   0.00000 × 10⁰   0.00000 × 10⁰ k = 4   0.00000 × 10⁰  0.00000 × 10⁰ k = 0, j = 10   0.00000 × 10⁰

TABLE 45 Example 6  Pupil Plane APR  N0 = N1 = 1 Local Coordinates x y zPosition 24.0000 −17.0000 0.0000 Vector VX −1.0000 0.0000 0.0000 VY0.0000 1.0000 0.0000 VZ 0.0000 0.0000 −1.0000 C0 0.000000 R 6.250000

TABLE 46 Example 6  Reflecting Surface S2  N0 = N1 = 1 Local Coordinatesx y z Position −2.5314 −20.8218 0.0000 Vector VX −0.9934 0.1144 0.0000VY 0.1144 0.9934 0.0000 VZ 0.0000 0.0000 −1.0000 C0 0.027834 ε A4 A6 A8A10 A12 1.0 −1.79016 × 10⁻⁴ −1.67583 × 10⁻⁶   1.28774 × 10⁻¹⁰ −1.95483 ×10⁻¹²   7.65360 × 10⁻¹⁵ Bjk j = 0 j = 1 j = 2 j = 3 j = 4 k = 0  0.00000 × 10⁰   0.00000 × 10⁰ −7.66855 × 10⁻³ −1.05589 × 10⁻⁴  1.80094 × 10⁻⁴ k = 2 −1.34309 × 10⁻³ −2.47669 × 10⁻⁴   3.64061 × 10⁻⁴−6.77025 × 10⁻⁷   5.06258 × 10⁻⁶ k = 4   1.89105 × 10⁻⁴ −4.18413 × 10⁻⁷  5.07642 × 10⁻⁶   0.00000 × 10⁰   0.00000 × 10⁰ k = 6   1.69143 × 10⁻⁶  0.00000 × 10⁰   0.00000 × 10⁰   0.00000 × 10⁰   0.00000 × 10⁰ k = 8  0.00000 × 10⁰   0.00000 × 10⁰   0.00000 × 10⁰ k = 10   0.00000 × 10⁰Bjk j = 5 j = 6 j = 7 j = 8 j = 9 k = 0 −2.20048 × 10⁻⁷   1.68175 × 10⁻⁶  0.00000 × 10⁰   0.00000 × 10⁰   0.00000 × 10⁰ k = 2   0.00000 × 10⁰  0.00000 × 10⁰   0.00000 × 10⁰   0.00000 × 10⁰ k = 4   0.00000 × 10⁰  0.00000 × 10⁰ k = 0, j = 10   0.00000 × 10⁰

TABLE 47 Example 6  Reflecting Surface S3  N0 = N1 = 1 Local Coordinatesx y z Position 120.1083 −68.6501 0.0000 Vector VX 0.9946 0.1034 0.0000VY −0.1034 0.9946 0.0000 VZ 0.0000 0.0000 1.0000 C0 0.004000 ε A4 A6 A8A10 A12 1.0   2.42154 × 10⁻⁷ −4.54612 × 10⁻¹¹ −4.53196 × 10⁻¹⁵   4.78069× 10⁻¹⁸   0.00000 × 10⁰ Bjk j = 0 j = 1 j = 2 j = 3 j = 4 k = 0  0.00000 × 10⁰   0.00000 × 10⁰ −3.19555 × 10⁻³ −4.89598 × 10⁻⁶ −3.50217× 10⁻⁷ k = 2 −3.47325 × 10⁻³   3.83920 × 10⁻⁶ −4.65460 × 10⁻⁷   1.42488× 10⁻⁹   1.44970 × 10⁻¹⁰ k = 4 −2.68538 × 10⁻⁷ −6.36563 × 10⁻¹¹  1.32216 × 10⁻¹⁰ −1.10135 × 10⁻¹³   2.68177 × 10⁻¹⁴ k = 6   4.52265 ×10⁻¹¹ −1.11849 × 10⁻¹⁴   1.82407 × 10⁻¹⁴   1.71914 × 10⁻¹⁸ −4.77890 ×10⁻¹⁷ k = 8   4.51594 × 10⁻¹⁵   7.81835 × 10⁻¹⁹ −2.39021 × 10⁻¹⁷ k = 10−4.77948 × 10⁻¹⁸ Bjk j = 5 j = 6 j = 7 j = 8 j = 9 k = 0 −6.21325 ×10⁻¹⁰   9.66229 × 10⁻¹¹   1.21715 × 10⁻¹² −2.21158 × 10⁻¹⁶ −2.25333 ×10⁻¹⁶ k = 2 −1.49391 × 10⁻¹³   1.59084 × 10⁻¹⁴ −8.78533 × 10⁻¹⁷ −2.49975× 10⁻¹⁷ k = 4   3.41538 × 10⁻¹⁷ −4.74824 × 10⁻¹⁷ k = 0, j = 10 −5.45701× 10⁻¹⁸

TABLE 48 Example 6  Reflecting Surface S4  N0 = N1 = 1 Local Coordinatesx y z Position 31.6335 −126.5920 0.0000 Vector VX −0.9635 0.2677 0.0000VY 0.2677 0.9635 0.0000 VZ 0.0000 0.0000 −1.0000 C0 −0.004044 ε A4 A6 A8A10 A12 1.0 −1.46748 × 10⁻⁸   6.91335 × 10⁻¹¹   5.22720 × 10⁻¹⁵ −7.41421× 10⁻¹⁸   0.00000 × 10⁰ Bjk j = 0 j = 1 j = 2 j = 3 j = 4 k = 0  0.00000 × 10⁰   0.00000 × 10⁰   2.83030 × 10⁻³   9.52598 × 10⁻⁶  1.08858 × 10⁻⁷ k = 2   5.08913 × 10⁻³   2.79153 × 10⁻⁵   3.96142 ×10⁻⁷   4.57770 × 10⁻⁹ −1.53563 × 10⁻¹⁰ k = 4 −3.26504 × 10⁻⁹ −1.04841 ×10⁻⁹ −2.34139 × 10⁻¹⁰ −6.24686 × 10⁻¹³ −3.88274 × 10⁻¹⁴ k = 6 −6.75734 ×10⁻¹¹   1.67018 × 10⁻¹⁴ −2.01313 × 10⁻¹⁴   2.54704 × 10⁻¹⁷   7.43461 ×10⁻¹⁷ k = 8 −5.29126 × 10⁻¹⁵ −1.89384 × 10⁻¹⁹   3.70810 × 10⁻¹⁷ k = 10  7.41814 × 10⁻¹⁸ Bjk j = 5 j = 6 j = 7 j = 8 j = 9 k = 0   4.29336 ×10⁻⁹   8.77554 × 10⁻¹¹   1.03950 × 10⁻¹² −4.68539 × 10⁻¹⁴ −7.83833 ×10⁻¹⁶ k = 2   6.92191 × 10⁻¹⁴ −3.08646 × 10⁻¹⁴ −1.39727 × 10⁻¹⁶  3.64635 × 10⁻¹⁷ k = 4 −4.94681 × 10⁻¹⁷   7.39562 × 10⁻¹⁷ k = 0, j = 10  3.50804 × 10⁻¹⁸

TABLE 49 Example 6  Reflecting Surface S5  N0 = N1 = 1 Local Coordinatesx y z Position 127.4805 0.0000 0.0000 Vector VX 1.0000 0.0000 0.0000 VY0.0000 1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C0 0.000000

TABLE 50 Example 6  Projection Surface S6  N0 = N1 = 1 Local Coordinatesx y z Position −17.5195 −626.5924 0.0000 Vector VX −1.0000 0.0000 0.0000VY 0.0000 −1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C0 0.000000

TABLE 51 Example 7  Display Surface S0  N0 = N1 = 1 Local Coordinates xy z Position 0.0000 0.0000 0.0000 Vector VX 1.0000 0.0000 0.0000 VY0.0000 1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C0 0.000000

TABLE 52 Example 7  Reflecting Surface S1  N0 = N1 = 1 Local Coordinatesx y z Position 75.4182 13.4971 0.0000 Vector VX 0.9475 0.3196 0.0000 VY−0.3196 0.9475 0.0000 VZ 0.0000 0.0000 1.0000 C0 −0.017525 ε A4 A6 A8A10 A12 1.0 −1.66382 × 10⁻⁷   2.82609 × 10⁻¹⁰   4.81274 × 10⁻¹⁵  1.10802 × 10⁻¹⁷   0.00000 × 10⁰ Bjk j = 0 j = 1 j = 2 j = 3 j = 4 k =0   0.00000 × 10⁰   0.00000 × 10⁰   3.73857 × 10⁻³ −5.01341 × 10⁻⁷  6.80585 × 10⁻⁷ k = 2   3.21110 × 10⁻³ −1.35376 × 10⁻⁶   1.40081 × 10⁻⁶  3.23581 × 10⁻⁹ −4.59668 × 10⁻¹⁰ k = 4   6.55625 × 10⁻⁷   1.80319 ×10⁻¹⁰ −5.12242 × 10⁻¹⁰   0.00000 × 10⁰   0.00000 × 10⁰ k = 6 −1.79598 ×10⁻¹⁰   0.00000 × 10⁰   0.00000 × 10⁰   0.00000 × 10⁰   0.00000 × 10⁰ k= 8   0.00000 × 10⁰   0.00000 × 10⁰   0.00000 × 10⁰ k = 10   0.00000 ×10⁰ Bjk j = 5 j = 6 j = 7 j = 8 j = 9 k = 0   2.63039 × 10⁻¹¹ −1.67688 ×10⁻¹⁰   0.00000 × 10⁰   0.00000 × 10⁰   0.00000 × 10⁰ k = 2   0.00000 ×10⁰   0.00000 × 10⁰   0.00000 × 10⁰   0.00000 × 10⁰ k = 4   0.00000 ×10⁰   0.00000 × 10⁰ k = 0, j = 10   0.00000 × 10⁰

TABLE 53 Example 7  Pupil Plane APR  N0 = N1 = 1 Local Coordinates x y zPosition 32.6886 −17.5000 0.0000 Vector VX −1.0000 0.0000 0.0000 VY0.0000 1.0000 0.0000 VZ 0.0000 0.0000 −1.0000 C0 0.000000 R 9.500000

TABLE 54 Example 7  Reflecting Surface S2  N0 = N1 = 1 Local Coordinatesx y z Position 3.2516 −26.2261 0.0000 Vector VX −1.0000 −0.0021 0.0000VY −0.0021 1.0000 0.0000 VZ 0.0000 0.0000 −1.0000 C0 0.029943 ε A4 A6 A8A10 A12 1.0 −6.67190 × 10⁻⁴ −5.76813 × 10⁻⁶   8.17928 × 10⁻¹¹ −3.16719 ×10⁻¹³   3.23298 × 10⁻¹⁶ Bjk j = 0 j = 1 j = 2 j = 3 j = 4 k = 0  0.00000 × 10⁰   0.00000 × 10⁰ −1.01265 × 10⁻² −5.73887 × 10⁻⁵  6.65231 × 10⁻⁴ k = 2 −4.67729 × 10⁻³ −8.88926 × 10⁻⁵   1.33180 × 10⁻³−2.21354 × 10⁻⁷   1.72808 × 10⁻⁵ k = 4   6.68079 × 10⁻⁴ −1.51675 × 10⁻⁷  1.72872 × 10⁻⁵   0.00000 × 10⁰   0.00000 × 10⁰ k = 6   5.76484 × 10⁻⁶  0.00000 × 10⁰   0.00000 × 10⁰   0.00000 × 10⁰   0.00000 × 10⁰ k = 8  0.00000 × 10⁰   0.00000 × 10⁰   0.00000 × 10⁰ k = 10   0.00000 × 10⁰Bjk j = 5 j = 6 j = 7 j = 8 j = 9 k = 0 −5.43254 × 10⁻⁸   5.75950 × 10⁻⁶  0.00000 × 10⁰   0.00000 × 10⁰   0.00000 × 10⁰ k = 2   0.00000 × 10⁰  0.00000 × 10⁰   0.00000 × 10⁰   0.00000 × 10⁰ k = 4   0.00000 × 10⁰  0.00000 × 10⁰ k = 0, j = 10   0.00000 × 10⁰

TABLE 55 Example 7  Reflecting Surface S3  N0 = N1 = 1 Local Coordinatesx y z Position 82.8882 −23.7811 0.0000 Vector VX 0.9984 −0.0558 0.0000VY 0.0558 0.9984 0.0000 VZ 0.0000 0.0000 1.0000 C0 0.008884 ε A4 A6 A8A10 A12 1.0   6.42863 × 10⁻⁷ −1.57431 × 10⁻¹⁰ −3.38299 × 10⁻¹⁴   2.89096× 10⁻¹⁸   0.00000 × 10⁰ Bjk j = 0 j = 1 j = 2 j = 3 j = 4 k = 0  0.00000 × 10⁰   0.00000 × 10⁰ −3.97653 × 10⁻³   8.09710 × 10⁻⁵  7.40631 × 10⁻⁷ k = 2 −6.38754 × 10⁻³   5.32223 × 10⁻⁵   9.56721 × 10⁻⁷  3.69191 × 10⁻⁸ −3.55872 × 10⁻¹⁰ k = 4 −6.26197 × 10⁻⁷ −7.40298 × 10⁻⁹−5.66599 × 10⁻¹⁰ −4.49197 × 10⁻¹¹ −7.08572 × 10⁻¹³ k = 6   1.00306 ×10⁻¹⁰ −6.82284 × 10⁻¹² −1.45392 × 10⁻¹³ −4.51260 × 10⁻¹⁵ −5.41853 ×10⁻¹⁷ k = 8   2.74914 × 10⁻¹⁴ −2.83504 × 10⁻¹⁶ −1.72951 × 10⁻¹⁷ k = 10−3.15292 × 10⁻¹⁸ Bjk j = 5 j = 6 j = 7 j = 8 j = 9 k = 0 −4.93872 × 10⁻⁹−6.96493 × 10⁻¹⁰ −1.86322 × 10⁻¹¹ −1.57288 × 10⁻¹³ −8.97245 × 10⁻¹⁶ k =2 −4.62424 × 10⁻¹¹ −6.99855 × 10⁻¹³ −6.99205 × 10⁻¹⁵ −3.75748 × 10⁻¹⁷ k= 4 −8.61053 × 10⁻¹⁵ −5.95656 × 10⁻¹⁷ k = 0, j = 10 −4.20010 × 10⁻¹⁸

TABLE 56 Example 7  Reflecting Surface S4  N0 = N1 = 1 Local Coordinatesx y z Position −32.1140 6.8008 0.0000 Vector VX −0.9873 0.1586 0.0000 VY0.1586 0.9873 0.0000 VZ 0.0000 0.0000 −1.0000 C0 −0.004801 ε A4 A6 A8A10 A12 1.0 −1.75892 × 10⁻⁶   2.16761 × 10⁻¹⁰ −1.41979 × 10⁻¹⁵ −1.02102× 10⁻¹⁹   3.17334 × 10⁻²⁴ Bjk j = 0 j = 1 j = 2 j = 3 j = 4 k = 0  0.00000 × 10⁰   0.00000 × 10⁰ −3.85943 × 10⁻² −1.19159 × 10⁻³ −8.95224× 10⁻⁶ k = 2   4.61352 × 10⁻³ −5.56763 × 10⁻⁴ −9.06592 × 10⁻⁶ −5.55578 ×10⁻⁸   4.86524 × 10⁻¹¹ k = 4 −2.66903 × 10⁻⁶ −4.63010 × 10⁻⁸   3.94905 ×10⁻¹⁰   1.39005 × 10⁻¹¹ −8.17065 × 10⁻¹⁴ k = 6   1.21750 × 10⁻⁹  4.54096 × 10⁻¹¹   5.40143 × 10⁻¹³   2.52351 × 10⁻¹⁵   3.52067 × 10⁻¹⁸k = 8 −4.64097 × 10⁻¹⁴ −9.49010 × 10⁻¹⁶ −5.28855 × 10⁻¹⁸ k = 10  5.83933 × 10⁻¹⁹ Bjk j = 5 j = 6 j = 7 j = 8 j = 9 k = 0   4.47919 ×10⁻⁸   1.44747 × 10⁻⁹   1.23245 × 10⁻¹¹   2.11007 × 10⁻¹⁴ −1.81745 ×10⁻¹⁶ k = 2   1.58714 × 10⁻¹² −1.17115 × 10⁻¹³ −1.25183 × 10⁻¹⁵ −3.66024× 10⁻¹⁸ k = 4 −2.06989 × 10⁻¹⁵ −8.04925 × 10⁻¹⁸ k = 0, j = 10 −6.94914 ×10⁻¹⁹

TABLE 57 Example 7  Reflecting Surface S5  N0 = N1 = 1 Local Coordinatesx y z Position 83.9399 0.0000 0.0000 Vector VX 1.0000 0.0000 0.0000 VY0.0000 1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C0 0.000000

TABLE 58 Example 7  Projection Surface S6  N0 = N1 = 1 Local Coordinatesx y z Position −41.0601 −477.3845 0.0000 Vector VX −1.0000 0.0000 0.0000VY 0.0000 −1.0000 0.0000 VZ 0.0000 0.0000 1.0000 C0 0.000000

TABLE 59 Example 8  Overall Size of Panel Display Surface I1 (mm):13.283 × 7.472 Size of Screen Surface I2 (mm): 1106 × 622 FNO inDirection of Longer Sides of Screen = 3.1 FNO in Direction of ShorterSides of Screen = 3.1

TABLE 60 Example 8  Surface Position and Rotation X Y Z X Y, Z SurfaceCoordinate Coordinate Coordinate Rotation Rotation Medium Panel DisplaySurface I1 0.000 680.462 −114.126 −24.938 0.000 First Mirror M1 Curved0.000 665.860 −73.268 −11.118 0.000 Air Second Mirror M2 Curved 0.000682.460 −254.319 −12.824 0.000 Air Third Mirror M3 Curved 0.000 616.446−4.826 −10.840 0.000 Air Fourth Mirror M4 Curved 0.000 341.827 −302.935−55.262 0.000 Air Fifth Mirror M5 Flat 0.000 217.746 8.580 8.596 0.000Air Sixth Mirror M6 Flat 0.000 1.802 −300.000 0.000 0.000 Air ScreenSurface (I2) 0.000 0.000 0.000 0.000 0.000 Air

TABLE 61 Example 8 Curved Surface Shape of First Mirror (M1) Radius ofCurvature (mm) = 106377.218 K = −9.439 × 10⁴ C(0,1) = 1.684 × 10⁻¹,C(2,0) = −2.258 × 10⁻³, C(0,2) = −2.369 × 10⁻³ C(2,1) = −9.083 × 10⁻⁶,C(0,3) = 1.096 × 10⁻⁵, C(4,0) = 8.012 × 10⁻⁹ C(2,2) = −4.285 × 10⁻⁸,C(0,4) = 2.231 × 10⁻⁸

TABLE 62 Example 8 Curved Surface Shape of Second Mirror (M2) Radius ofCurvature (mm) = 267.910 K = −4.304 × 10⁻¹ C(0,1) = 4.674 × 10⁻², C(2,0)= −1.645 × 10⁻⁴, C(0,2) = −1.742 × 10⁻⁴ C(2,1) = 6.569 × 10⁻⁸, C(0,3) =1.008 × 10⁻⁶, C(4,0) = 7.508 × 10⁻¹⁰ C(2,2) = −2.565 × 10⁻¹⁰, C(0,4) =−2.336 × 10⁻⁹

TABLE 63 Example 8 Curved Surface Shape of Third Mirror (M3) Radius ofCurvature (mm) = 570.148 K = −3.109 × 10² C(0,1) = 7.535 × 10⁻², C(2,0)= 5.066 × 10⁻⁴, C(0,2) = 1.925 × 10⁻³ C(2,1) = −1.293 × 10⁻⁵, C(0,3) =−6.269 × 10⁻⁶, C(4,0) = 2.246 × 10⁻⁷ C(2,2) = 4.088 × 10⁻⁷, C(0,4) =5.380 × 10⁻⁸

TABLE 64 Example 8 Curved Surface Shape of Fourth Mirror (M4) Radius ofCurvature (mm) = −993.286 K = 2.645 × 10 C(0,1) = −1.238, C(2,0) =−1.032 × 10⁻³, C(0,2) = 1.167 × 10⁻³ C(2,1) = 1.086 × 10⁻⁵, C(0,3) =6.456 × 10⁻⁶, C(4,0) = 1.133 × 10⁻⁹ C(2,2) = −1.795 × 10⁻⁷, C(0,4) =−4.084 × 10⁻⁸

TABLE 65 Example 9 Overall Size of Panel Display Surface I1 (mm): 26.624× 19.968 Size of Screen Surface I2 (mm): 1024 × 768 FNO in Direction ofLonger Sides of Screen = 3.0 FNO in Direction of Shorter Sides of Screen= 4.5

TABLE 66 Example 9 Surface Position and Rotation X Y Z X Y, Z SurfaceCoordinate Coordinate Coordinate Rotation Rotation Medium Panel DisplaySurface I1 0.000 1225.125 −127.470 −34.830 0.000 Air First Mirror M1Curved 0.000 1168.810 −111.456 −24.720 0.000 Air Aperture Stop ST 0.0001235.920 −285.524 −25.369 0.000 Air Second Mirror M2 Curved 0.0001242.861 −287.234 −26.051 0.000 Air Third Mirror M3 Curved 0.0001136.690 −60.687 −24.388 0.000 Air Fourth Mirror M4 Curved 0.000 817.349−408.232 −81.766 0.000 Air Fifth Mirror M5 Flat 0.000 256.510 −42.517−2.937 0.000 Air Sixth Mirror M6 Flat 0.000 −5.416 −300.000 0.000 0.000Air Screen Surface (I2) 0.000 0.000 0.000 0.000 0.000

TABLE 67 Example 9 Curved Surface Shape of First Mirror (M1) Radius ofCurvature (mm) = 47670.865 K = −6.434 × 10⁴ C(0,1) = 2.113 × 10⁻¹,C(2,0) = −9.706 × 10⁻⁴, C(0,2) = −1.863 × 10⁻³ C(2,1) = 7.277 × 10⁻⁶,C(0,3) = 5.779 × 10⁻⁶, C(4,0) = 1.856 × 10⁻⁸ C(2,2) = 1.133 × 10⁻⁷,C(0,4) = 9.931 × 10⁻⁸

TABLE 68 Example 9 Curved Surface Shape of Second Mirror (M2) Radius ofCurvature (mm) = 266.862 K = −9.462 × 10⁻¹ C(0,1) = −1.537 × 10⁻²,C(2,0) = −1.508 × 10⁻⁴, C(0,2) = −1.841 × 10⁻⁴ C(2,1) = 7.947 × 10⁻⁸,C(0,3) = 2.548 × 10⁻⁷, C(4,0) = 5.091 × 10⁻⁹ C(2,2) = 9.101 × 10⁻⁹,C(0,4) = 3.779 × 10⁻⁹

TABLE 69 Example 9 Curved Surface Shape of Third Mirror (M3) Radius ofCurvature (mm) = 511.833 K = −1.186 × 10² C(0,1) = 1.197 × 10⁻¹, C(2,0)= 6.380 × 10⁻⁴, C(0,2) = 1.251 × 10⁻³ C(2,1) = −4.052 × 10⁻⁶, C(0,3) =−3.980 × 10⁻⁶, C(4,0) = 1.494 × 10⁻⁷ C(2,2) = 2.385 × 10⁻⁷, C(0,4) =6.638 × 10⁻⁸

TABLE 70 Example 9 Curved Surface Shape of Fourth Mirror (M4) Radius ofCurvature (mm) = 0.000 K = −1.879 × 10² C(0,1) = −2.058, C(2,0) = −8.631× 10⁻⁴, C(0,2) = 1.908 × 10⁻³ C(2,1) = 5.097 × 10⁻⁶, C(0,3) = 6.368 ×10⁻⁶, C(4,0) = 2.201 × 10⁻⁹ C(2,2) = −5.440 × 10⁻⁸, C(0,4) = −2.417 ×10⁻⁸

TABLE 71 Example 10 Overall Size of Panel Display Surface I1 (mm):13.283 × 7.472 Size of Screen Surface I2 (mm): 1106 × 622 FNO inDirection of Longer Sides of Screen = 3.1 FNO in Direction of ShorterSides of Screen = 3.1

TABLE 72 Example 10 Surface Position and Rotation X Y Z X Y, Z SurfaceCoordinate Coordinate Coordinate Rotation Rotation Medium Panel DisplaySurface I1 0.000 429.000 217.364 −72.761 0.000 Air First Mirror M1Curved 0.000 379.122 202.375 −61.047 0.000 Air Second Mirror M2 Curved0.000 529.372 98.361 −62.009 0.000 Air Aperture Stop ST 0.000 535.873120.360 −63.273 0.000 Air Third Mirror M3 Curved 0.000 290.291 211.413−61.724 0.000 Air Fourth Mirror M4 Curved 0.000 297.822 −109.328−105.696 0.000 Air Fifth Mirror M5 Flat 0.000 −451.659 −322.158 −105.1090.000 Air Sixth Mirror M6 Flat 0.000 −715.000 250.000 0.000 0.000 AirScreen Surface (I2) 0.000 0.000 0.000 0.000 0.000

TABLE 73 Example 10 Curved Surface Shape of First Mirror (M1) Radius ofCurvature (mm) = 55105.558 K = −9.416 × 10⁴ C(0,1) = 2.665 × 10⁻¹,C(2,0) = −1.488 × 10⁻³, C(0,2) = −2.314 × 10⁻³ C(2,1) = 7.735 × 10⁻⁶,C(0,3) = 1.046 × 10⁻⁵, C(4,0) = 8.701 × 10⁻⁹ C(2,2) = 1.657 × 10⁻⁷,C(0,4) = 8.682 × 10⁻⁸,

TABLE 74 Example 10 Curved Surface Shape of Second Mirror (M2) Radius ofCurvature (mm) = 266.893 K = 5.482 × 10⁻¹ C(0,1) = 3.794 × 10⁻⁴, C(2,0)= −1.355 × 10⁻⁴, C(0,2) = −1.900 × 10⁻⁴ C(2,1) = 3.553 × 10⁻⁷, C(0,3) =6.314 × 10⁻⁷, C(4,0) = −5.225 × 10⁻⁹ C(2,2) = −1.127 × 10⁻⁸, C(0,4) =−6.737 × 10⁻⁹

TABLE 75 Example 10 Curved Surface Shape of Third Mirror (M3) Radius ofCurvature (mm) = 669.724 K = −3.507 × 10² C(0,1) = 9.594 × 10⁻², C(2,0)= 9.897 × 10⁻⁴, C(0,2) = 1.845 × 10⁻³ C(2,1) = −3.058 × 10⁻⁶, C(0,3) =−6.449 × 10⁻⁶, C(4,0) = 2.237 × 10⁻⁷ C(2,2) = 4.923 × 10⁻⁷, C(0,4) =1.248 × 10⁻⁷

TABLE 76 Example 10 Curved Surface Shape of Fourth Mirror (M4) Radius ofCurvature (mm) = −1508.718 K = 6.332 × 10 C(0,1) = −1.224, C(2,0) =−4.739 × 10⁻⁴, C(0,2) = 1.439 × 10⁻³ C(2,1) = −3.909 × 10⁻⁷, C(0,3) =4.994 × 10⁻⁶, C(4,0) = −2.171 × 10⁻⁸ C(2,2) = −1.135 × 10⁻⁷, C(0,4) =−2.627 × 10⁻⁸

TABLE 77 Examples 8 to 10 Values of Conditional Formulae, etc. DL HLDL/HL θ Example 8 1000 622 1.61 38.07 Example 9 1394 768 1.82 50.85Example 10 1650 622 2.65 55.68

What is claimed is:
 1. An oblique projection optical system for leadingrays of light from a display surface on which an image is displayed to aprojection surface in such a way that a ray of light from a center ofthe display surface is obliquely incident on the projection surface inorder to project a magnified image of the image displayed on the displaysurface onto the projection surface, the oblique projection opticalsystem including a plurality of reflecting surfaces having a power,wherein at least two of the reflecting surfaces having a power have afree-form curved surface, and, of all the reflecting surfaces having apower, the one closest to the projection surface has a negative powerand at least one of the others has a positive power; and wherein thedisplay surface has a smaller dimension in a height direction than in awidth direction, the reflecting surfaces having a power each reflect therays of light from the display surface in such a way as to deflect therays of light in the height direction of the display surface, a pupilplane is located between the one of the reflecting surfaces having apower that is second-closest to the display surface and the displaysurface, and the following conditions are fulfilled: Fnoy≧Fnoz,Fnoy≦4.5, and Fnoz≦4.0, where Fnoy represents an f-number in a directioncorresponding to the height direction of the display surface, and Fnozrepresents an f-number in a direction corresponding to the widthdirection of the display surface.
 2. An oblique projection opticalsystem as claimed in claim 1, wherein there are four of the reflectingsurfaces having a power, which have, in order from a display surfaceside, a positive, a negative, a positive, and a negative power.
 3. Anoblique projection optical system for leading rays of light from adisplay surface on which an image is displayed to a projection surfacein such a way that a ray of light from a center of the display surfaceis obliquely incident on the projection surface in order to project amagnified image of the image displayed on the display surface onto theprojection surface, the oblique projection optical system including aplurality of reflecting surfaces having a power, wherein at least two ofthe reflecting surfaces having a power have a free-form curved surface,and, of all the reflecting surfaces having a power, the one closest tothe projection surface has a negative power and at least one of theothers has a positive power, and wherein the display surface has asmaller dimension in a height direction than in a width direction, thereflecting surfaces having a power each reflect the rays of light fromthe display surface in such a way as to deflect the rays of light in theheight direction of the display surface, and the following condition isfulfilled: D/H≧0.35, where H represents a dimension of the projectionsurface in a direction corresponding to the height direction of thedisplay surface, and D represents a maximum length, in a directionnormal to the projection surface, of a space through which the rays oflight pass to travel from the display surface to the projection surface.4. An oblique projection optical system as claimed in claim 3, whereinthe following condition is fulfilled: 30≦β≦100, where β represents aratio of a size of the projection surface to a size of the displaysurface.
 5. A rear projection optical system comprising a projectionoptical system for projecting an image displayed on a panel displaysurface onto a screen surface, wherein the projection optical systemincludes at least four curved-surface reflecting mirrors, and wherein,assuming that a ray of light that travels from a center of the paneldisplay surface through a center of an aperture stop to a center of thescreen surface is called a screen center ray, the following condition isfulfilled: 0.5<DL/HL<3.5, and 10°<θ<70°, where DL represents an opticaldistance traveled by the screen center ray from a last surface of theprojection optical system to the screen surface, HL represents adimension of the screen surface in a direction parallel to a planeformed at the center of the screen surface by a normal to the screensurface and the screen center ray, and θ represents an angle ofincidence at which the screen center ray is incident on the screensurface.
 6. A rear projection optical system as claimed in claim 5,wherein the projection optical system forms no intermediary image in anoptical path from the panel display surface to the screen surface.
 7. Arear projection optical system as claimed in claim 5, wherein, of thecurved-surface reflecting mirrors, at least the one closest and the onesecond-closest to the panel display surface have a substrate made ofglass.
 8. An oblique projection optical system as claimed in claim 3,wherein there are four of the reflecting surfaces having a power, whichhave, in order from a display surface side, a positive, a negative, apositive, and a negative power.
 9. A rear projection optical systemcomprising a projection optical system for projecting an image displayedon a panel display surface onto a screen surface, wherein the projectionoptical system includes at least four curved-surface reflecting mirrors,and wherein at least three of the curved-surface reflecting mirrors havea free-form curved surface.
 10. An oblique projection optical system forleading rays of light from a display surface on which an image isdisplayed to a projection surface in such a way that a ray of light froma center of the display surface is obliquely incident on the projectionsurface in order to project a magnified image of the image displayed onthe display surface onto the projection surface, the oblique projectionoptical system including a plurality of reflecting surfaces having apower, wherein at least two of the reflecting surfaces having a powerhave a free-form curved surface, and, of all the reflecting surfaceshaving a power, the one closest to the projection surface has a negativepower and at least one of the others has a positive power, and whereinthere are four of the reflecting surfaces having a power, which have, inorder from a display surface side, a positive, a negative, a positive,and a negative power.